Thought Experiments


Math: Discovered, Invented, or Both?

“The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

Eugene Wigner wrote these words in his 1960 article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The Nobel prize-winning physicist’s report still captures the uncanny ability of mathematics not only to describe and explain, but to predict phenomena in the physical world.

Credit: Flickr user Barney Livingston, adapted under a Creative Commons license.

How is it possible that all the phenomena observed in classical electricity and magnetism can be explained by means of just four mathematical equations? Moreover, physicist James Clerk Maxwell (after whom those four equations of electromagnetism are named) showed in 1864 that the equations predicted that varying electric or magnetic fields should generate certain propagating waves. These waves—the familiar electromagnetic waves (which include light, radio waves, x-rays, etc.)—were eventually detected by the German physicist Heinrich Hertz in a series of experiments conducted in the late 1880s.

And if that is not enough, the modern mathematical theory which describes how light and matter interact, known as quantum electrodynamics (QED), is even more astonishing. In 2010 a group of physicists at Harvard University determined the magnetic moment of the electron (which measures how strongly the electron interacts with a magnetic field) to a precision of less than one part in a trillion. Calculations of the electron’s magnetic moment based on QED reached about the same precision and the two results agree! What is it that gives mathematics such incredible power?

The puzzle of the power of mathematics is in fact even more complex than the above examples from electromagnetism might suggest. There are actually two facets to the “unreasonable effectiveness,” one that I call active and another that I dub passive. The active facet refers to the fact that when scientists attempt to light their way through the labyrinth of natural phenomena, they use mathematics as their torch. In other words, at least some of the laws of nature are formulated in directly applicable mathematical terms. The mathematical entities, relations, and equations used in those laws were developed for a specific application. Newton, for instance, formulated the branch of mathematics known as calculus because he needed this tool for capturing motion and change, breaking them up into tiny frame-by-frame sequences. Similarly, string theorists today often develop the mathematical machinery they need.

Passive effectiveness, on the other hand, refers to cases in which mathematicians developed abstract branches of mathematics with absolutely no applications in mind; yet decades, or sometimes centuries later, physicists discovered that those theories provided necessary mathematical underpinnings for physical phenomena. Examples of passive effectiveness abound. Mathematician Bernhard Riemann, for example, discussed in the 1850s new types of geometries that you would encounter on surfaces curved like a sphere or a saddle (instead of the flat plane geometry that we learn in school). Then, when Einstein formulated his theory of General Relativity (in 1915), Riemann’s geometries turned out to be precisely the tool he needed!

At the core of this math mystery lies another argument that mathematicians, philosophers, and, most recently, cognitive scientists have had for a long time: Is math an invention of the human brain? Or does math exist in some abstract world, with humans merely discovering its truths? The debate about this question continues to rage today.

Personally, I believe that by asking simply whether mathematics is discovered or invented, we forget the possibility that mathematics is an intricate combination of inventions and discoveries. Indeed, I posit that humans invent the mathematical concepts—numbers, shapes, sets, lines, and so on—by abstracting them from the world around them. They then go on to discover the complex connections among the concepts that they had invented; these are the so-called theorems of mathematics.

I must admit that I do not know the full, compelling answer to the question of what is it that gives mathematics its stupendous powers. That remains a mystery.

Go Deeper
Editor’s picks for further reading

NOVA: The Great Math Mystery
Is math invented by humans, or is it the language of the universe? NOVA takes on this question in a new film premiering April 15, 2015 at 9pm on most PBS stations.

NOVA: Describing Nature with Math
How do scientists use mathematics to define reality? And why? Peter Tyson investigates two millennia of mathematical discovery.

The Washington Post: The Structure of Everything
Learn more about the “unreasonable effectiveness of mathematics” in this review of Mario Livio’s book “Is God a Mathematician?”

Tell us what you think on Twitter, Facebook, or email.

Mario Livio

    Dr. Mario Livio is an internationally renowned astrophysicist, a best-selling author, and a popular speaker. He is a fellow of the American Association for the Advancement of Science. He has published more than 400 scientific papers on topics ranging from dark energy and cosmology to black holes and extrasolar planets. Dr Livio is also the author of five popular science books, including "The Golden Ratio" (an International Bestseller for which he received the "Peano Prize" and the "International Pythagoras Prize") and "Is God A Mathematician?” — which was the basis for the 2016 Emmy-nominated NOVA program “The Great Math Mystery." Livio's most recent book, "Brilliant Blunders," was a national bestseller in the U.S., and was selected by the Washington Post as one of the "Best Books of the Year.” His upcoming book, "WHY? What Makes Us Curious" will appear in July 2017.

    • As you said, Newton invented the calculus in order to explain the phenomena he saw. In other words, mathematics is simply a way to express regularities. The question may be why there are regularities in the universe. But if one assumes that there are, then mathematics is simply a way to express them once we start to see them. It’s less a mystery than this post makes it.

      Put another way, if we encounter and begin to recognize a regularity for which no mathematics exists, we invent the mathematics necessary to formalize what we see. It’s not that mathematics is independent of the regularities we want to express. It’s our (invented) languages for expressing them.

      • Michael Keating

        I agree. It is the description of relationships between objects or phenomena. Certain ratios appear repeatedly in nature because the rules affect all or most things equally. This seems intuitive to me. Is it possible that a different base numbering system could demystify or untangle complicated equations?

    • Jess Tauber

      Several years ago I discovered that atomic shell structures (both electronic and nuclear) appeared to depend on mathematical combinatorics that came right out of the Pascal Triangle. For example, in the simple harmonic oscillator model of shell structure in the spherical atomic nucleus, magic numbers of nucleons (where shells are filled, analogously to noble gases in electronic structure) are 2,8,20,40,70,112,168- these are exactly doubled tetrahedral numbers 1,4,10,20,35,56,84… Variations on this theme also help determine the harmonic oscillator magic numbers for ellipsoidally deformed nuclei. Differences between the spherical magics are single copies of doubled triangular numbers, and the oscillator ratio of a sphere is 1:1. When the oscillator ratio is 2:1 then TWO copies of each double triangular number sum to give each next deformed magic in succession. When the ratio is 3:1, then THREE copies so sum. On the other hand, for oblately deformed nuclei, the relation is divisional rather than multiplicational. For the sphere, a double triangular number difference exists for every magic number. But for the oblate nucleus of oscillator ratio 1:2, that difference is between EVERY OTHER magic, and for ratio 1:3 between EVERY THIRD magic and so on. Exceptions exist when you haven’t yet accumulated enough magics (equal to the denominator of the oscillator ratio), and then the magics are simple double triangular numbers until you reach that point. No exceptions. More complicated ‘realistic’ models of the nucleus include a spin-orbit interaction as well as modifications to the potential well the nucleons feel. But even here there are Pascal relationships. So-called ‘intruder’ levels (the highest spin orbital-partials from the next higher shell) lower their energies (because of the spin-orbit coupling) enough to drop into the previous shell, adding their number of nucleons to the magics (so 40+10 (g9/2)=50, 70+12 (h11/2)=82, 112+14(i13/2)=126, 168+16(j15/2)=184 and so on. The DEPTH of such penetration (the total count just before or after one adds the intruder nucleons) turns out (for neutrons) to be exactly doubled triangular numbers. So 40-2=38, with g9/2 starting just after, and 50-2=48, where the 10 nucleons terminate. 70-6=64, with h11/2 starting after, and 82-6=76, where the 12 nucleons terminate. And so on. There are many other Pascal mathematical relationships present in nuclear shell structure. The reason they work seems to be that in the harmonic oscillator nucleus each period is composed of sums of every other orbital, segregated by parity. So s=2, p=6, ds=12, fp=20, gds=30, hfp=42, igds=56, jhfp=72… Note that these are all doubled triangular numbers. Further, the SIZES of the additions given by intruders aren’t arbitrary, but are exactly those needed to increase the total size of the period they add to to the next double triangular number- 20+10=30, 30+12=42, 42+14=56 and so on. The phenomenon preserves double triangularity!

      In the electronic structure, if we reformat the periodic table so that the s-block elements are on the right edge (the ‘left-step’ table of Charles Janet (discovered in the late 1920’s, which organizes things the way a physicist would, rather than a chemist), then the alkaline earths (but also including helium) terminate each period. If you look at the atomic numbers of every other s2 element they are ALL tetrahedral numbers (4,20,56, *120). And all the intermediate atomic numbers are the arithmetic means of these (so (0+4)/2=2, (4+20)/2=12, (20+56)/2=38, (56+120)=88… This ‘triad’ relationship was discovered originally in the early 19th century but based on atomic weights rather than atomic numbers, and was the basis of the periodic relation’s discovery (most famously by Mendeleev but he wasn’t the only one). Because of these numerical relationships the entire periodic system can be reconceptualized in three dimensions as a perfect tetrahedron composed of close-packed spheres, each sphere representing one element. As with the nuclear system there are many other Pascal relationships in shell structure. Each electronic period is a half/double square number in size: 2,8,18,32,50… But in the Janet table they are all paired for size, which gives sums of duals as full squares 4,16,36,64… And if you sum THESE you get the tetrahedral numbers 4,20,56,120. The question is what are we to make of all this- is it mere ‘numerology’ or can we gain some advance to our understanding of Nature through such studies?

      • Samuel Bass

        Look up tetryonics. Sounds like you’ve got a piece of the puzzle figured out.

      • Electron    ---∗---

        An electron is shaped like the metal spines of an umbrella (without the hinges or fabric of course).
        One string extents from where your hand would hold it up to the center of axis. There, eighteen strings (or radii) extent out in the same curved disc type shape as the umbrella. The last string goes straight up (the same length as all the rest) and connects with the field in space (space is made of the same stuff by the way).

        Notice the way some elements in vertical columns in the Periodic table chart have an atomic number with difference of 18 between them. Most of the chart is like that (notice how many columns there are).

        It’s because 18 is the determinant number in electron shell configuration.

        Every particle starts with radii (strings) that are arranged in the dodecahedral axis shape
        That’s the vertices of the dodecahedron or the faces of the icosahedron (platonic solids.)
        This is a way stuff can form and happen automatically.

        Every electron particle has 20 strings.
        One string is attached to the proton.
        One string connects with space (or an electron in the next outer shell).
        The other 18 strings form the electron disc.
        When electrons connect with each other they have 18 strings to play with.

        Check the larger noble gases: Argon 18, Krypton 36, Xenon 54, Radon 86, the amount of electrons in outermost shells will always sum to 18, the first three even have atomic numbers that are multiples of eighteen. Three groups of six radii from one electron can form (along with seven other electrons) the corners of a cube or the "Octet Rule" and seal off the package.

    • Both Invented & Discovered…I’m with Mario. The intelligence is in the geometry itself, if it’s self-consistent, it can also be self-organising.
      My @Quora answer to “Do the Fibonacci Sequence, Pi, and the Golden Ratio do anything to prove that the universe was intelligently designed?”…

    • Jess Tauber

      One other point. On a whim I took Fibonacci numbers and mapped them as ATOMIC NUMBERS. It turned out, for at least all known elements, they mapped as FIRST/LEFTMOST elements in the orbital half-rows. This is relevant because electrons fill lobes singly before they start to double up, so the left orbital half has only singly filled lobes and the right orbital half starts doubling lobes. AND, the odd Fib numbers were always mapped to the singly filled first/left half orbital, and the even Fib numbers to the double-filling second/right half orbital. At 144 shell models predict that this mapping scheme will be severely out of alignment, but then again we’ll probably never see such heavy elements. Interestingly, the related Lucas numbers 2,1,3,4,7,11,18,29,47,76,123…. tend to map to LAST/rightmost positions in orbital half rows. This works up to 18, but afterwards misaligns. Even so Nature seems to have behavioral fixes: 29, copper, and 47, silver, are in the same column (coinage metals) with electronic configurations that are ‘anomalous’. By position in the table they *should* have d9,s2 but instead have d10,s1, where one electron has been abstracted from a full s shell (rendering it half filled but that is ok for the Lucas trend) and filling the d shell, again ok in the Lucas system). 76, osmium, has the correct electronic configuration for its position within the periodic table, but acts ‘as if’ it were xenon in some of its compounds, that is a noble gas with a full orbital. Apparently d6 is reinterpreted as if it were p6.

    • George Gantz

      With all due respect, mathematics is a conprehensive and integrated whole arising at the point of conscious awareness of the first distinction of something from nothing through a priori laws of logic. This is the thesis of my essay The Hole at the Center of Creation, one of 200 submitted in the 2015 FQXi essay contest on this topic. Anyone seriously interested in the wide and disparate views on this issue should visit Public comments and essay ratings are welcome!

    • Moliehi Marake

      For so long, I wondered why most principles and studies in life are better understood when expressed and explained in mathematical terms, but this piece has just given some perspective to my questions. I personally consider mathematics an innate part of man, just as language is. It is a tool within us that naturally exists, a tool we resort to in dire need to answer some of the phenomena around us.

    • Paul Stamison

      I just saw the program and was I ever so excited as I love math and I always viewed it just like the program did. I will try to get the DVD and will show it to my after school program for some brilliant grade 7 and 8 students. Thank you.

    • Enrique Vela

      First of all, there’s no mystery in the usefullness of maths to understand nature because maths are simply the way our brain works to make calculations and our brain is evolutionary adapted to the realities with which it has to deal. Maths work because they are an adaptation to nature just like the eye to the sunlight.
      And second, it’s a myth that the adaptation of maths to nature is so good. When it comes down to reality, almost no mathematical model à la Maxwell equations really explain the whole of it. They are idealizations good to understand and to express but they always need corrections to account for whole phenomena and finally we end up trying to solve series of infinite diferential equations that work well for some purposes (QED and the magnetic moment of the electro) but no so well for others. Here is the limit, probably evolutionary limit, of the mathematical approach, when we concoct an extremely complex tapestry of infinite equations as our best guess from our theoretical first principles but not even then can get results except for some well defined problems where the approach works.
      It’s time to avoid falling in these kind of ideas that are so cheerfully propagated by the theoretical scientists and mathematicians and return to the humble non-pretencious concept of the maths as an instrument whose abstract content, in the best of cases, reveals only the inner workings of our brain, not of nature.

    • low pressure is moving

      your presentation make many statements that are false. You should never compose a declaratory sentence that is not true. For example the notion that gravitational attraction by itself keeps the planets in place is false. Also, are you sure that the constants in the equations are truly constants in all media, time (if it exists) and expanse?

    • bfast

      Is mathematics discovered or invented? Wrong question! The real question is, “Is invention discovery or creation”?

      While there is a creative component in invention — the consideration to attempt to produce a certain something, there is also a fundamental discovery component to invention: figuring out how to do the something that creativity imagined.

      As a multiply-patented inventor, I frequently hear people tell me “what would be nice is …” This is human creativity at work. However, just because Hollywood-style hover-boards would be nice, making such a thing is not possible without an (as yet unmet) discovery.

      Consider Thomas Edison working to invent the light bulb. He is famous for claiming that he tried about 1,000 things that didn’t work before finally finding a method that did work. Why didn’t these other methods work? Oh yea, because though they were surely creative, they did not fit within the framework of reality. And when Edison finally found something that worked, what did he utter? “Eurika, I found it!”? Probably something like that. His utterance was very similar to what a gold prospector utters when he discovers the mother lode. His utterance is similar, his methods — strategic searching — is similar as well.

      The answer to the question “is mathematics discovery or invention” then, is “Yes!”

    • Teodor Jovanovski

      Well it was all okay until….

    • Mongo

      Math is oracular. Amazing.

      • and I quote:

        “Nevermind that S#!^, here comes Mongo!”

        “Mongo only pawn in game of life…”

        I simply had to add both of these quotes as no one else had yet…

        As for the article, math was discovered, observed, recorded, derived, and invented. The fact that it was always there makes the work no less meaningful as our society would not exist as we know it without our knowledge of these fundamental mathematics.

    • Korrazon Cold

      Everything is formed by a surrounding 4pi Spherical inward absorption +1=0 now -1 outward emission of electromagnetic waves.. . .All motion is 2pi Spiral.. . .And all 3D directions are 4pi Spherically curved thus Wave Centres of photon energy are everywhere relative to an infinite future of potential possibilities continuously coming into +1=0 now -1 out of existence.. . .

      There exists a profound and perfect 4pi Spherical symmetry between positive and negative electric charge compressing energy input +1=0 now -1 de-compressing when that symmetry was broken forming spiral symmetry.. . .Now we get the imperfect 2pi Spiral symmetry of the Fibonacci sequence.. . .From Spiral radians 360 degrees, to Spirals of elements, DNA, Seashells, Spiral galaxies, and even living cells.. . . Simply because generation of any information exceeds radiation during the first half of the cycle +1=0 now -1 radiation exceeds generation during second half of the cycle as the constant outward momentum of electromagnetic radiation repels like charged particles absorbing energy input +1=0 now -1 emitting the density from the two previous vectors spiralling out the Fibonacci sequence seen almost everywhere in nature.. . .

      The fact that this spherical region of the one infinite universe has a limited observable range as the light waves come from a distance radius. . ..Absorbing that information now is observable in the shape of an inverse sphere. . ..4pi R2.. . .And the fact that the electron is a perfectly round vibrating sphere geometrically are two sides of the same + now – coin!!.. . .

      There exists only two combinations of these Two Spherical Sine Wavefronts multiplying inward +1=0 now -1 dividing outward at right angles from their sources.. . .They have opposite vectors and quantum spin forming the positron +1=0 now -1 electron Wave Centre.. . .4pi R2=/N pi Re2.. . .Or any dipole moment!!

      Now space is a division of Solidity into entropy as time unfolds C2 the second law of thermodynamics.. . .But also in the future E2 will equal a multiplication of Volume +1=0 now at the expense of gravitational potential -1.. . .E2=M2 C4+P2 C2.. . .

      • Sean Nanoman

        Can’t figure out why so many of you post this EXACT same piece of gobbly gook which is neither a response to anyone or making any clear statement whatsoever. For example would you please explain in introductory terms what a ‘4pi inward spherical absorption’ is? Or would you at least explain why this would not apply to a 3pi system, or with adsorption as opposed to absorption?
        Much appreciated.

    • Yes, math is great but you could have two people with completely opposite / opposing theories. The maths for both (theories) could be peered reviewed and deemed absolutely stinkin’ correct.
      But only one of the theories can be correct so that means one of the theories must be wrong.

      They actually both might be wrong.

      So if you prove something with math you actually DID NOT prove or disprove anything.

    • antonio carlos motta

      The speed of light is constant into of severa spacetime cpntinuitu és. The deformations of the curvatures of spacetime implica in diferentes topological geometrias that does the securvatures bbê variável with the spacetime that determini a vale tô speed of light.the riemann ian in 4 dimensional manifold are generated thr ou ghosts of torsion tensor fields calculei by the métrica of the noncommutative that are the fluxions of newton, with the curvatures bê generated by the fluxions o pose that me a sue the the micro topological geometry with securageometry

    • John TAnkerssley

      Mathematics is, I think, quite simply the language of the universe, the language of God. It is also the language of morals and reality. Please see my bog at

    • IdPnSD

      “… we forget the possibility that mathematics is an intricate combination of inventions and discoveries”.

      Mathematics is completely false. Real numbers are not objects of nature, because it neither grows on trees nor it can be mined from earth. Since real numbers are not objects of nature, mathematics has never been discovered. Discovery means something that existed in nature. Real numbers did not exist in nature. One law of nature says – nonexistent cannot become existent.

      Take a look at chapter one on truth at the blog site and in other chapters for many examples of false math and physics, including Newton’s first law, Quantum Mechanics etc. After all you cannot create anything truth using something that is false like real numbers.