Infinity Is a Beautiful Concept – And It’s Ruining Physics

By Max Tegmark | February 20, 2015 9:00 am


I was seduced by infinity at an early age. Georg Cantor’s diagonality proof that some infinities are bigger than others mesmerized me, and his infinite hierarchy of infinities blew my mind. The assumption that something truly infinite exists in nature underlies every physics course I’ve ever taught at MIT—and, indeed, all of modern physics. But it’s an untested assumption, which begs the question: Is it actually true?

A Crisis in Physics

There are in fact two separate assumptions: “infinitely big” and “infinitely small.” By infinitely big, I mean that space can have infinite volume, that time can continue forever, and that there can be infinitely many physical objects. By infinitely small, I mean the continuum—the idea that even a liter of space contains an infinite number of points, that space can be stretched out indefinitely without anything bad happening, and that there are quantities in nature that can vary continuously.

The two assumptions are closely related, because inflation, the most popular explanation of our Big Bang, can create an infinite volume by stretching continuous space indefinitely. The theory of inflation has been spectacularly successful and is a leading contender for a Nobel Prize. It explains how a subatomic speck of matter transformed into a massive Big Bang, creating a huge, flat, uniform universe, with tiny density fluctuations that eventually grew into today’s galaxies and cosmic large-scale structure—all in beautiful agreement with precision measurements from experiments such as the Planck and the BICEP2 experiments. But by predicting that space isn’t just big but truly infinite, inflation has also brought about the so-called measure problem, which I view as the greatest crisis facing modern physics.

Physics is all about predicting the future from the past, but inflation seems to sabotage this. When we try to predict the probability that something particular will happen, inflation always gives the same useless answer: infinity divided by infinity. The problem is that whatever experiment you make, inflation predicts there will be infinitely many copies of you, far away in our infinite space, obtaining each physically possible outcome; and despite years of teeth-grinding in the cosmology community, no consensus has emerged on how to extract sensible answers from these infinities. So, strictly speaking, we physicists can no longer predict anything at all!

This means that today’s best theories need a major shakeup by retiring an incorrect assumption. Which one? Here’s my prime suspect: ∞.

Infinity Doesn’t Exist

A rubber band can’t be stretched indefinitely, because although it seems smooth and continuous, that’s merely a convenient approximation. It’s really made of atoms, and if you stretch it too far, it snaps. If we similarly retire the idea that space itself is an infinitely stretchy continuum, then a big snap of sorts stops inflation from producing an infinitely big space and the measure problem goes away. Without the infinitely small, inflation can’t make the infinitely big, so you get rid of both infinities in one fell swoop—together with many other problems plaguing modern physics, such as infinitely dense black-hole singularities and infinities popping up when we try to quantize gravity.

In the past, many venerable mathematicians were skeptical of infinity and the continuum. The legendary Carl Friedrich Gauss denied that anything infinite really exists, saying “Infinity is merely a way of speaking” and “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics.” In the past century, however, infinity has become mathematically mainstream, and most physicists and mathematicians have become so enamored with infinity that they rarely question it. Why? Basically, because infinity is an extremely convenient approximation for which we haven’t discovered convenient alternatives.

Consider, for example, the air in front of you. Keeping track of the positions and speeds of octillions of atoms would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum—a smooth substance that has a density, pressure, and velocity at each point—you’ll find that this idealized air obeys a beautifully simple equation explaining almost everything we care about: how to build airplanes, how we hear them with sound waves, how to make weather forecasts, and so forth. Yet despite all that convenience, air of course isn’t truly continuous. I think it’s the same way for space, time, and all the other building blocks of our physical world.

We Don’t Need the Infinite

Let’s face it: Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable universe contains only about 1089 objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about seventeen decimal places. Yet real numbers, with their infinitely many decimals, have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics. We describe even a single bit of quantum information (qubit) using two real numbers involving infinitely many decimals.

Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.

Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it.

edge book Excerpted from This Idea Must Die, edited by John Brockman. Used with permission.


Image by Mark Ramsay via Flickr

CATEGORIZED UNDER: Space & Physics, Top Posts
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  • Rich

    Its not a concept, its reality.

    • atanu nath

      How do you know for sure ? Because in Physics we don’t understand Infinity as infinity… we understand in a limiting sense.. something tending to infinity.. something that is much much greater than the scale of our theory or experiment. We never deal with exact infinity.. because they create havoc in the loop level of QFT calculations…

    • Jeff Kesner

      You can’t claim a monopoly on the meaning of the word “reality” in a metaphysical philosophical discussion.

  • LEK56

    As a lay-person the one question that has always bugged me (also adding related questions) is this: If the universe is infinite, it must have infinite mass. Then why isn’t it solid? So the universe isn’t infinite? I KNOW I am showing my ignorance on this, so be gentle.

    • 7eggert

      Hilbert’s infinite Hotel: Infinite (mass) particles arrive, each one gets a room. 99 infinite groups of space-between arrive and want a room, too. Each particle will move to (old room number * 100).

      Now room 1 .. 99 are rent to space, and room 100 to a particle.

  • Uncle Al

    Vast mind-years rigorously derive hectares of non-empirical physical theory. Euclid had Bolyai, Newton had relativity and quantum mechanics, particle theory had Yang and Lee. A founding postulate is testably defective. Theory cannot falsify its origins. One must experiment. How does physics stubbornly fail?

    Massless boson photons detect no vacuum refraction, dispersion, dissipation, dichroism, or gyrotropy. Postulate exact vacuum isotropy toward fermionic matter (quarks, hadrons). Parity violations, symmetry breakings, chiral anomalies, baryogenesis, Chern-Simons repair of Einstein-Hilbert action are vacuum trace chiral anisotropy acting only upon hadrons.

    Measure vacuum trace chiral anisotropy, DOI: 10.5281/zenodo.15445 Chiral Eötvös experiment, chiral microwave rotational temperature, chiral enthalpy of fusion, chiral pawnbroker rotation. LOOK.

    Quick shot! Noether’s theorems couple exact vacuum isotropy with angular momentum conservation. Vacuum anisotropy leaks Milgrom acceleration. “Dark matter” curve-fits the Tully-Fisher relation.

    • Jeff Kesner

      Fwiw I think every one of those “failures” is because particle theory assumes they are talking about one particle when in fact experiment shows that quarks always appear in pairs, never separately.

  • Ryan Reece

    that “1089 objects” should be 10^{89}

    • Keln

      Well, depending on your definition of “object”, there could be only 1,089 of them.

      • Ramome

        Or just one depending on your perspective.

  • Phillip Rulon

    Well done. I’ve thought for many years that what we know about extreme physics is influenced by more than a little hubris. We know quite a bit about the universe up to energy densities of about 10^14 Kelvin. Along the road to that understanding new physics has emerged about every 3 orders of magnitude. Modern cosmology seems to assert that our current understanding can be extrapolated to 10^30 Kelvin or higher without any new physics to complicate the theory. Historically this is unsupported. It’s quite likely that new physics will be discovered before we understand 10^18 Kelvin.

  • Henry Massey

    I thought it was generally accepted that space-time was not continuous, but granular at the Planck Length/Planck Time scales.

    • 7eggert

      We are still searching for directly observable evidence – AFAIK.

      Currently you can’t e.g. explain light without e=hf (solar cells depend on the photoelectric effect), you can mathematically prove that a position can’t be more precise than a Planck length, but you can’t see the effects predicted by assuming space to be made from small cells either – yet. I think they didn’t reach enough precision while looking for the effect?

  • 7eggert

    Dividing infinite by infinite – my first thought was using L’Hopital’s rule.

    • minstrelmike

      Calculus is the art of dividing zero by zero.

      • GuestWhom

        if zero multiplied by anything equals zero, then 0/0 = anything


        • 7eggert

          And yet you can get meaningful results.

          Plot f(x)=(x²+2x)/x.
          Calculate lim x->0 f(x).

          • TheCarMan

            Well, technically you don’t divide by ZERO, you merely are looking for the value approached by your function as X approaches ZERO. Even your function (which DOES approach a real number) is not defined at X = 0.

          • 7eggert

            Yes, you can insist on having the problem and not seeing a way to solve it.

          • TheCarMan

            And just where did I imply it couldn’t be solved? Did you actually READ what I posted?

            “Even your function (which DOES approach a real number) is not defined at X = 0.”

            In this case, one doesn’t even need to use L’Hopital’s Rule. Just look at it. As x approaches 0, x² becomes negligible compared to 2x., hence x² + 2x approaches 2x. And 2x / x approaches 2. DUH !

          • 7eggert

            Sorry for not posting the whole literature on calculus within a simple sentence. Also sorry for using examples!!!

        • daqu

          You’re right: 0/0 can equal any number at all. (That’s why it is often called “indeterminate” rather than “meaningless”.)

          • diogenes

            A = B – assume
            A^2 = AB – multiply by the same quality
            A^2 – B^2 = AB – B:^2 – subtract the same quantity
            (A-B)(A+B) = (A-B)B – factor both sized
            A + B = B – after dividing by (A-B) **
            B + B = B – substitute B for A
            2B = B – combine
            2=1 – divide by B

            ** Since A = B, dividing by A-B is dividing by zero – which is prohibited.
            Unwitting Division by zero symbolically has fooled many a manipulator into believing false spurious result.

            If we started with (A-B)(A+B) = (A-B)B,
            we simplify getting A^2-B^2 = AB – B^2
            simplify again to A^2 = AB (adding B^2 to both sides)
            Then assume A is not zero, and divide by A,
            concluding A=B except when A is zero.
            The straight line A=B has a hole in it at (A,B)=(0,0)
            We can take the limit of B/A as A approaches zero, then the limit of B as a approaches zero is zero, but that is a different branch of mathematics than simple division.

        • diogenes

          Division by zero is formally undefined. It is not a valid mathematical statement. As such, any statement using it is false, and as good logicians, know, anything follows from false.

          • bt

            That’s why infintesimals exist – they aren’t 0.

    • Franck Merlot

      Infinite is not a number, it’s a concept.

      • diogenes

        The axiom of infinity states. there exists a smallest number (Aleph Null) such that if N is number less than Aleph Null, then N+1 is also less than Aleph Null.

        Aleph Null is the first trans-finite number. When Achilles reaches the tortoise, he has reached the number Aleph Null as the step in his race. As he speeds past the tortoise, more transfinite numbers are required for the successive “mile markers”.

        • Franck Merlot

          As you said, it’s an axiom. Axioms can be defined arbitrarily and even if they are logically valid they might not be logically sound.

          “All pink unicorns are pink” is a logically valid statement but it is not useful in describing reality.

          • diogenes

            “Valid” is a property of an argument, not of an axiom, (except when you equivocate ‘valid’ and ‘true’ – a lay use of the term valid.)
            An argument is ‘valid’ if and only if assumed to be true can not lead to a false conclusion.

    • PG

      Why would that not be 1.

    • PG

      Why would that not be 1. Is that too basic

      • 7eggert

        Look at the formula I posted, plot the graph and you’ll see.

        • PG

          Why would I need any formula when division is all that is needed. No matter what infinity may be divided by itself will equal 1.

          • 7eggert

            Just in order to see why you are wrong.

          • diogenes

            Have you ever tried to get a computer or calculator to divide by zero? Note, there is no ‘infinity’ key on calculators.

            Like 0/0, infinity/infinity is undefined, and limits produce something else. Limits find points that make the relation continuous which is otherwise discontinuous at a point.

          • PG

            why are we talking about zero, whatever you ascribe to infinity as the value is the same as you ascribe to the divisor, this is very basic mathematics. So even though you cannot ascribe a value to infinity it is never zero so division by itself would always equate to 1.

  • Mark Byron

    A number of concepts depend on what it’s limit is as it approaches infinity, like the number e, which is (1+1/n)^n as n heads to infinity; it comes in handy for continuous compounding and a number of calculus applications. As an irrational number, you can’t calculate it exactly, but can come very close to whatever degree of accuracy you need.

    The universe might not be truly infinite nor might we be able to get infinitely small units of time to affect continuous compounding, but those assumptions make for some more-elegant and useful models.

    • Jeff Kesner

      fwiw I think the universe ticks. [because of Planck Length / Planck Time], however this is in philosophical conflict with relativity theory which uses continuous equations (i.e. infinite)

  • P. J. Dillon

    The universe isn’t a rubber band though can produce enough material to make an infinite amount of them. Infinity is akin to the geometric shape of the universe. To limit the universe by removing infinity limits the imagination that conceived of what is around us in the first place. If infinity doesn’t exist, you have to prove that it doesn’t, not speculate and remove it without evidence, especially when everything studied and measured so far is finite in comparison.

    • Franck Merlot

      This is an argument from ignorance. There is no evidence that infinity physically exists. It is merely a man-made concept for the time being. The burden of proof solely remains on those who assume infinity as real. It is fallacious to dodge the burden of proof on those who simply point out that there is no evidence. The “no evidence” part is a fact, not a claim.

  • Andrew Howard

    The use of infinitesimals in basic Calculus would be a candidate surely. It has always niggled me: f'(x) is Lim( h -> 0) f(x)…what would happen if the 0 was replaced with Planck length for physical equations?

    • 7eggert

      It would be impossible to tell the difference between 0 and any value less than the Planck length. Also it would be impossible to tell the difference between any value between 0 and the Planck length from the Planck length itself. “Lim” would be useless.

      Physics as we know it is undetermined for these small values.

      PS: I’m searching for a source to really learn SRT/ART and quantum mechanics. Most sites either stop after giving you an overview or start after you supposedly have learned a lot beyond classical mathematics and school physics, using symbols in a way completely different from what I know.

      • Andrew Howard

        When you integrate using lim(h->0) it allows you to eliminate many terms of the equation. This “trick” is probably the lowest level where infinities, or infinitesimals are introduced into Physics. If h is in fact not quite zero then this may have interesting implications for things like gravity at large scales…We know that spacial grain is not zero but Planck length. Although very small it is not zero, and very small things add up over cosmic scales.
        We are after all looking for a way for more insight into unifying the micro (QM) and macro (GR), so ensuring the tools (maths) make no “simplifying” assumptions (like h->0) seems like a good candidate.
        I don’t really have the math to explore the concept (I learned Lagrangians and partial differentiation via economics, not physics!), but I like the idea that “Dark matter” and “Dark energy” may be explained by a rational tweak to the maths.

        • 7eggert

          lim x->const requires that you can use any converging sequence of x_0, x_1,… and get the same result.

          In quantum physics, you don’t get the same result, so you can’t use lim. At least from my understanding.

          For dark matter, energy, I suspect them to be mathematical artifacts, just like Newtonian equations can’t predict Mercury’s orbit. Just like Einstein had to accept c=const in order to create SRT and inertia mass = gravitational mass to create ART, there will be something simple that forces us to create a new theory.

  • gtrmath

    Most high school age students are first exposed to the concept of infinity with the infinite geometric series 1/2+1/4+1/8+1/16+1/32+… = 1. Then there are power series expansions for pi and e, functions with asymptotes, and improper integrals. Infinity isn’t going away anytime soon…

  • silver fox

    brockman: the last of one the really great minds in any hemisphere. The Edge. He cannot be circumscribed in praise or diminished by criticism. thank you for many years of joy and education.

  • David S. Warren

    To “beg the question” does not mean “begs us to ask some question,” but rather, that the question assumes the answer. Not that I quite understand the rest of this language either.

  • Juror #8

    Well, can’t these scientists just fake the data like they do on everything else?

    • GridUser

      Brian Williams already tried that. They caught him. I think that’s what that was about.

      • gtrmath

        Brian Williams counted to infinity, twice.
        Brian Williams trisected an angle, with his bare hands.

    • DarkMatterExplainer

      Hurrrrr. If you were smart enough you could do your own experiments to (try and) prove anything in established science wrong. One of the main tenets of science is FALSIFIABILITY, dumbass.

      • Juror #8

        But faking it is so much easier and gets you published and praised by your peers.

        No one really checks the data, so why bother?

  • Mike 3/505

    Here’s my layman’s question…If the Universe isn’t infinite, what’s the edge/end of it look like? How do we know when we’ve arrived there?

    • Juror #8

      There is a door. You open it and go into a white room with thousands of televisions in it and talk with a man in a white suit with a gray beard.

      He’ll speak in parables. You’ll get up and leave more confused than you were when you went in.

      • Mike 3/505

        And when I try to combine a leap in the air and multiple kicks, I fall on my a$$.

      • Gary Ronan

        It must be Colonal Sanders! I was wondering where he went!!

    • 7eggert

      The observable edge is the universe 380 000 years after the Big Bang when it became transparent after being opaque.

      We can’t arrive there, we are always at the center of the universe.

      • Mike 3/505

        We can’t arrive there, we are always at the center of the universe.

        Reminds me of my ex wife.

        • RedStatePatriot


      • MaximumOvertroll

        Well, the center of OUR universe. Anything at any point in the universe appears to be at the center of its own universe from its reference frame.

        • Dingo

          There are certain patrons what implicates what is the centre. It doesn’t is anything at any point. A spinning wheel has just one centre, if you are in the middle or outer right or left on it, the middle is still the middle. The universe is to big to know but our galaxy is a good example.

          • MaximumOvertroll

            Take a spinning wheel the size of the universe, pick an arbitrary point on the wheel, using that point as the center, draw a circle big enough that an observer in the center cannot observe anything beyond the circumference. Now draw circle of equal size directly adjacet to the first. Observers at the center of both circles would assume they are at the center of their own separate universes evem though they are neghbors within the same universe nowhere near its center.

          • Dingo

            The universe is a giant. Milky ways are beautiful. Our galaxy is one of many. Earth and the sun are not in the middle of our milky way, in contrary, more on the outside. There are no pictures of the whole milky way. But study shows it has a spiral shape and exists outta planets and stars. I do not know if they believe there are more milky ways but we should suppose they do because other wise they would not believe in other life somewhere…
            But like always, there will be many faults.
            The fact is, they can’t believe in infinity and they can’t believe in nothing. And the funny part is, they want you to believe in something but they don’t know what because it would be to complicated. Well, just as nothing is infinity a definition. And if you claim the definition is wrong, you should change the definition or you should claim a new word for your new definition.
            Almost pulling some Catholics with it but gonna shut up about the Pope. They do believe nothing and describe a touch of it in the bible, but of course it is again a definition and definition are made by… Humans!
            So supposing the univers is ever expanding, you probably are right there is no middle of universe but universe is not milky way! The middle of a spinning wheel is the middle, right? But the subject was infinity ? Interpretation is a game we suppose..

        • Andy

          That’s a philosophical assumption, not empirical science. Observationally, we _appear_ to be near the “center” of the universe. I’m going to infer from those observations . . . that we actually are.

          • MaximumOvertroll

            As would any observer assume they are the center of their universe from their position in it. It’s not philosophical, it’s reletivity.

          • Anechidna

            We know our system is in one of the spirals towards the end of the spiral and that our Galaxy is one of many.

            If observational we appear to be near the centre of the universe is that simply because we are looking around the outer curve of the universe which ever way we look so appear to be in the centre even if we were off to one side of it. Is the outer edge of the universe the curve of infinity but because we are looking at it in from two directions we fail to see that we have reached the end of the universe. For example think of driving the car down the road in one direction and then if we reverse our travel we see things we didn’t see on the outbound trip and we miss things we saw outbound as we travel back.

        • 7eggert

          It’s the same universe, watched by different observers.

          • MaximumOvertroll

            Well tell that to the observers too far away to see our visible universe. For all intents and purposes, they appear to be in another universe

          • 7eggert

            If it eventually touches, it’s the same universe. They’ll just have to wait.

          • MaximumOvertroll

            A supermassive blackhole can contain a galaxy’s worth of information, that information is inaccessable to the universe it resides in even though they are “touching”. It might as well be its own seperate universe.

          • VendicarDecarian0

            Be careful of your use of the term “contains”. since as viewed by an observer “outside” the event horizon, nothing ever manages to fall under the event horizon.

            So there may be no inside at all.

            I prefer the view that there is another universe inside every black hole. But that is just personal preference.

          • Jason

            Sorry, evidently i have no idea what i’m doing here.

      • protn7

        that’s it’s edge in time that’s visible in all directions at once.

        • Dingo

          Are there any other milky ways in our universe? If there are we should be afraid to be eaten by a bigger one. The unanswered question is.. From were does all these living and non living material come? From big bang, OK, but how did the big bang gathered his material to exploid? A football blowed outta the field by some football player? Home run would we say! What a blow! These is an infinity search, and that is why they give up! Infinity exists even for researchers.
          (But Catholics know the answer: God!, Hurry up become a catholic and become a crusader and conquer the world!!) We could assume he who tells infinity doesn’t exists is probably to much catholic!
          The knowledge to talk about infinity within math is not present but know what it is about and the graphs to.

          • protn7


    • Jeff Kesner

      The universe isn’t infinite because the laws of physics are uniform, that is to say finite in description but thought to be infinite in scope.

      • Dingo

        @jeff. That is something they are not sure about. There are 4 possibilities that are being proved by x radiation ray’s that are possible. These 4 are mentioned by the Belgian George Lemaitre. Now there is a possibility among those 4 that says our galaxy could shrimp again but the other 3 are more likely to.. a bunch of people. And even if our galaxy comes back together somehow, it doesn’t implicate it hasn’t moved in the universe. If we exist outta matter there could be somewhere else happening the same, with or without life. And these are all law of physics who are telling these! (OK, or some guy called George Lemaitre in this case)

    • Aeffesstoo

      ” How do we know when we’ve arrived there?”

      You fall off the edge and dragons eat you.

      • the truth

        You can’t know. If that were the end of the universe than when you got there you would no longer exist. So you would not know.

    • joshuagenes

      You will know when you run out of space : )

      • Yoikes

        And how would you know anyway. “All dressed up and no place to go”!

        • joshuagenes

          I know all the secrets of the Universe just ask me.

    • Michael Ryan

      first keep in mind its never infinite yet its still expanding and faster than light so we could never get there anyway. Even if we could we couldnt know where it is because that would involve measurement and measurement or “space” is relative to objects but if objects are constantly moving in a constantly expanding space how can anything be measured.Oh did I mention time gets warped so even if you could predict the velocities perfectly to get the three space coordinates to fix a position you couldn’t get the fourth dimension you couldn’t even be sure you were even in the first four dimensions anymore. google the dimensions after four if you really want a headache..but to answer your question, nothing the nothing beyond the universe is not a vacuum or an emptiness a void etc it simply doesnt exist so you couldn’t observe it you cant approach it your approaching it expands the universe but not into it because it isnt I think its a harder concept than infinity, THERE IS NOTHING BEYOND THE UNIVERSE THATS WHY WE CALL IT THE UNIVERSE.As the matter in the universe gets farther away from each other the forces that bind them weaken gravity and inertia die the universe dies entropy death

    • delbarco

      a circle does not have infinite length, yet it has no end. Might the universe be something like that?

      • nfojunky

        A four-dimensional hypershpere would do this.

        From a song by Modest Mouse: “The universe is shaped exactly like the earth – if you go straight along enough you’ll end up where you were.”

        That’s always been my idea of the shape of the universe.

    • modmans2ndcoming

      It can be closed (finite) yet the surface (dimensions) we inhabit can be be without borders. Think about how the earth is a finite but you can walk in any direction without walking off the earth.

    • Robert Anweiler

      You simply can’t prove infinity, because you are a finite being with a finite mind and a finite amount of time to exist, and since infinity would be bigger than the finite, the concept itself is simply unfathomable in the human mind. It is an impossible-to-prove concept, therefore we have reached what literally cannot be known by humanity.

      • diogenes

        Mathematicians play with a host of transfinite numbers. All you need is Piano’s successor axiom and the axiom of infinity. Take a look at Cantor’s diagonal argument showing that there are more real numbers than can be counted by the.

    • Nate

      The answer is there is no edge – not in the way we would typically imagine, anyway!

      Rip off a stream of toilet paper and lay it out flat. Now, pick up the two ends, twist one end once, then attach them both together.

      Trace a line with your finger along the toilet paper. If you loop around once, you’ll end up on the opposite side of the sheet from where you started. But if you continue tracing, eventually you’ll end up exactly where you started.

      This makes perfect sense to us, of course. We can see the twist and loop that causes this to happen. But a 2D being would be immensely confused by the idea that if you travel far enough in one single direction, you could eventually reach your starting point.

      The idea of space curving in more directions than the three dimensions we perceive confuses us just the same. But that’s exactly what it does (or so we believe). We aren’t sure of its shape yet, but regardless, we will probably not find an “edge” to the universe, because the real boundaries lie in higher dimensions.

      • Mike 3/505

        Whenever I think this stuff…the sheer vastness of the universe, I wonder how we can even have half the effrontery the human race has, when we try to play God with some of the stuff we do. One of these days, we are really gonna do something really stupid.

        • VendicarDecarian0

          Bio-warfare / Bio-Terrorism.

    • ericlipps

      Suppose the universe isn’t infinite but exists on the three-dimensional (four-dimensional, counting time) surface of a higher-dimensional object–say, a hypersphere. In that case, even though limited in size, it would have no “edge” or “end.” If you could travel far enough, you’d end up returning to your starting point from the opposite direction.

      • VendicarDecarian0

        Sorry. That doesn’t follow.

        Presume that the observer is stationary in space.

        Unless they are not stationary in time as well, then as the observer moves through time, it would see the edge of the hyper-sphere changing it’s distance.

        If we start from the center and move outward, then we would observe the universe to be contracting.

        We would never wrap back on ourselves.

    • Dan Poisson

      According to Mr Einstein you can’t reach the edge because there isn’t any. What will happen if you travel long enough is you will end up back where you started.

      • VendicarDecarian0

        Not necessarily.

    • diogenes

      The definition of “universe” is that it includes everything. To assume it has an “edge” violates the definition, because an edge separates inside from outside, but nothing can be “outside” the universe. That’s why we call the universe “unbounded” If you walk around in a circular tube with a completely featureless surface, you can never tell when you get back to where you started; you can not find an end – Remember, it’s featureless. We can not see far enough to see into the period in the past when the universe was small enough, and our mathematical theory is insufficient to predict the physical details of that period. That’s why the farther back in time and space we conjecture, the more possible theories of what might be arise. String theory, Branes, multiverse, etc.

    • VendicarDecarian0

      The edge is indeterminate. It’s location depends on your velocity for one thing, and since space is full of undulating fields the edge is undulating at the plank scale as well.

      Try as you might, you will never reach the edge of the universe, since it is always the same distance from you.

    • Andrey Lalev

      Not sure if you still follow this, but you should watch “No Edge: The Shape of the Universe” by Zogg from Betelgeuse – it explains your question perfectly

    • Russ Hamilton

      The edge of it does not look like anything and you can’t know when you have arrived because you can’t be there. If there is a boundary to the universe it is a boundary of time. Without time as reference point, existence is impossible. So yeah it is tough to get your head around but it could be just that.

      • diogenes

        Universe, by definition, means all that exists, If the universe had an edge, it would separate what is in the universe from what is outside the universe, but since the universe means to include all that there is, there can’t be anything “outside the universe, so there can not be an edge, because an edge describes what is inside from what is across the edge. You can draw a line around the equator of the earth and call it an edge, but both halves are on the earth. Like the circle on the sphere, the 3 dimensions of the universe curve in time. Time is inside the universe, so not only is there no place outside the universe there is also no time outside the universe.

  • jhs

    Infinity is settled science.

    • Andy

      The belief that infinity is a “settled science” is a faith-position, not empirical.

      • jhs

        like man made climate change?

        • DarkMatterExplainer

          Yeah, except that doesn’t happen.

        • VendicarDecarian0

          Sorry boy. You have been watching too much Alex Jones nonsense.

          • jhs

            I’m not your boy. IPCC email scandal.

      • jhs

        Hard to prove infinity, I will grant you that.

    • VendicarDecarian0

      No. Infinity is settled mathematics.

      Mathematics need not be applicable 1:1 with reality.

  • Keln

    My understanding of infinity is that it is simply something used for mathematical purposes. It doesn’t actually describe anything real and isn’t meant to. The only “real” infinite thing is numbers. If you could live forever, you could count forever without end. But I don’t think anyone truly believe the universe is “infinite”. So infinity belongs only to mathematics, where it is useful.

    • 7eggert

      In order to count forever, you have to remember the current number. One day the number will be too large, the universe will be too small to hold the information contained in that single number.

      • Keln

        Good point. I hadn’t considered that.

      • Jeff Kesner

        I don’t think so, you can always invent new notation to suit your needs, like for example the tower function. After a while this gets dull, so “oo” is just easier to write and discuss.

        • 7eggert

          No matter what you do, the number contains an entropy. This is the amount of information you need to hold that number.

          You can use different encodings, but to remember which encoding you used, you need entropy, and using a different encoding will just move around the entropy between the numbers you encountered. There is no way to count to 1024 without using 10 bit of entropy.

          • waffleironmarch

            But if infinity is true, then the numbers can be made infinitely small regardless of the currwnt size of the universe. Infinity is potential, which is what exists prior to physical expansions. Things don’t have to resemble easily relatable physical analogies to exist. Concepts may be intangible but they show themselves as the emergent interactions between things that can be detected, and are measured with numbers. There is perceptible reality and abstract reality, and without either of these things, the other is meaningless, like science and philosophy.

          • 7eggert

            Each number can get as small as you like, but the total “size” of all numbers needs to be as big as the total entropy of all numbers, because you need to distinguish them.

      • Dan Poisson


    • albertobladerunner

      You just don’t know, like everyone else here. I do not know it is even possible to know the answers to these questions.

  • Mrowley

    It’s the infinitely small concept that blows my mind. I mean, doesn’t everything have to made of smaller somethings for matter to exist? If not, then at some subatomic level something simply, self-evidently “is”.

    Obviously I’m no physicist.

    • Jeff Kesner

      Actually that’s a problem in philosophy. You are speaking in “materialistic” terms, which you can notice by your use of the word matter.

      • wikiwhat

        But philosophically, does it matter?

        • Jeff Kesner

          Matter does it, philosophically. –Yoda

          • Steve

            “Philosophy” is the new “Physics” …plus they both start with pH which is, obviously a picohenry.

    • Dingo

      Sometimes it is better to believe in nothing. Atheism is following the good pad! :-) It means space creates his own univers and beyond that universe there is nothing. But when space arrives, universe gets expanded. Now.. Maybe there are giants who are following the univers experiment and they find it to interesting to break it down because planet earth seems to have some curious activity. But if it is we will not know because we are to small for these to see or discover and the giant is looking at us. We probably are not dangerous viruses for these giants. And we do not know how they occurs, but they probably know the concept: nothing. But a human needs something because they do not understand how nothing can exist. There needs and there luxury comes all from something! Nothing it is not possible! And there is were your god comes in, to comfort you and to let you live the way you wanne live without worrying about nothing. (because if you meet nothing, you run really really fast, because it is like dead himself is running after you) But maybe we should assume you could just believe in nothing and that would be enough but it probably would not be interesting enough for common folks and only well disciplined and educated people would find a drive to go further and look further. That is the reason to believe in something (not necessarily a god) and your community would have a drive to. The other option is to be ‘just not interested’ and it would save you a whole lot of time and thinkering effort. Though, personal would be preferred to be the disciplined person, so you should could be not interested, but if you like you could. You could do handcrafted or more educated jobs. Anyhow.. my age is going and my age, created by these universe, isn’t letting me be disciplined or skilled anymore.. So that is the reason it is sad youth infinity doesn’t exist.

    • Jason

      “I” am.

    • VendicarDecarian0

      Is matter infinitely divisible?

      We know it isn’t.

      The question is, “Is space infinitely divisible?”.

      The answer to that is also no. (probably), since space is created by virtual energy, and virtual energy is quantized.

      Space also then becomes quantized.

  • bt

    The finite universe and the infinite universe co-exist ,both unable to perceive the other.

  • seescaper

    We have to give up the concept of a “particle.” What is that? What size or density is it? What is it made of? No answers. Particles are miniature singularities. So what do we replace the particle with? The standing wave concept (Wave Structure of Matter) of Dr. Milo Wolf. Google “beyond the point particle.” Essentially a “particle” is in reality composed of a standing wave structure formed by the rotation of an incoming wavefront that is rotated at the center to produce an outgoing wavefront. The in waves come from all other wave-centers in that wave-center’s observable universe. The outwave contributes to all other particle’s inwaves. All are interconnected. This explains Mach’s principle, gravity, spin, charge, “red-shift” all elegantly and unifies quantum theory and relativity, all of which can be derived from this.

    • Jeff Kesner

      The concept of physics is usually just “the math works”, in which case, I’m willing to go one step further and say that a particle is some form of math, if not in fact, just a vector. This works best with the “wave function” which seems to apply to electrons as well as photons.

  • CrazyHungarian

    So the universe is a little less than infinite? Maybe infinity – 1?

  • johnwerneken

    Fascinating article and discussion!

  • JoeThePimpernel

    Infinity ain’t what it used to be.

  • fedup

    This article is typical of “science” in the modern world. “We don’t know diddly so let’s just decide it doesn’t matter”.

  • putupjob

    we are finite beings and cannot understand infinity.

  • Charles R. Kiss

    I think the relevant issues regarding infinity and cosmology, is that one can have an infinite time, and a finite space, or vice versa. Equally, there are functions that allow the observer to perceive these quantities as such when in fact they may be either open or closed.

    Unfortunately, these functions are not linear.

    Understandably, scientists seem to have a preference for functions that are linear without any basis for them other than for simplicity -even though it just complicates everything and/or leads them to other conundrums, or impossibilities.

  • Dan

    The author is quite correct. Infinity is a mathematical concept, but not a physical one. Avoiding it will make mathematical descriptions of the physical world slightly more cumbersome, but infinitely more accurate!!

    • Charles R. Kiss

      Not that simple, you can have limits that are closed but one can only approach the limit infinitely closely, ie. asymptotes.

  • Bee Farms

    there is no such thing as any of this physical matter or energy or forces etc etc. We just THINK there is something “physical” — there are only “ideas” — nothing else is really there.

  • f_this_state

    Replace infinity with 0 and this article could come right out antiquity. I’ll play devil’s advocate here and pretend to be a person dead set against the abstract idea of nothingness as would have existed in antiquity.

    0 doesn’t exist. It’s not provable. The idea that nothing could exist is a logical fallacy. How can you have nothing? If you did have nothing, how would you measure it? You can’t test for 0. This new idea of nothing is ridiculous and it’s hurting out quest for answers. We have to look past this abstract concept of nothingness and concentrate on the reality of the world around us.

    0 doesn’t exist. We don’t need 0.

    • YeahRight

      0 is a symbol which simply means that a set is empty. All empty sets are identical. That’s trivial and has a well defined ontology. I put five coins on the table and take five coins away. The result of those two combined operations is called zero, BY DEFINITION.

      The only fallacy in your example is the idea that one can put a universe on the table and then take the same universe away. That is not a possible physical operation because of the definition of universe (which, mathematically speaking, is NOT a set).

      • Franck Merlot

        Is a set something or nothing?

    • Jason

      Lovely point… Absence of absence…

    • Franck Merlot

      That’s exactly the point. Non-existence is inherently paradoxical.

  • blair houghton

    Horseshit. ” Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite.”

    Feynman once talked about the deviation from simulation that reality will take for even the simplest systems, for instance a single electron making its way around its probability amplitude cloud about the nucleus of its atom, that will cause it to be completely unpredictable very shortly after we start the simulation and the real system.

    No computation that we can construct will ever approach the precision of reality. Thus we can never construct a simulation to model the entire universe. We can make gross predictions with large error bars, and we can use real observation to refine our models, but our models will always at best be cartoon versions of the real thing.

    Luckily, reality likes cartoons, so things like F=ma and g=GMm/r^2 never stop being funny.

    • YeahRight

      Reality doesn’t have a precision. Only measurements do, but then, measurements are not the same as reality. Go back to Plato and his shadows on the cave wall.

  • Douglas J. Bender

    Doesn’t General (or Special, or both) Relativity depend upon assuming that space (or space/time) is continuous?

    • YeahRight

      No, it doesn’t. It only relies on the assumption being sufficiently true for the scale on which general relativity has been tested, which is the case.

  • YeahRight

    You have failed completely to understand physics. Physics is a series of approximations. Each approximation is supposed to cover a sufficiently large number of observations of reality. Many approximations use mathematical concepts that are more expressive than what they are being used to describe naively. Of course, when we make a measurement of e.g. a voltage, we don’t actually assign a real number with infinite precision to that measurement. We assign a probability to the hypothesis that the next measurement of the same voltage will lie within a certain interval of integer multiples of the resolution of our instrument. It’s simply much easier to abstract that to a real number than to talk all the time about probability distributions of digital readouts while we are holding a voltmeter in our hands.

    The same is true for any other measurement that is done correctly. Curiously experimental physicists usually have a much better handle on the ontology of physical measurements than theoreticians. We actually know what they mean while the theory folks are often all wrapped up in mathematical structures that are merely simplifications of real metrology procedures.

  • Matthew

    Infinity isn’t ruining anything… Its a necessity for the theory of everything. If you set infinity in stone, then present the big bang as the point when the physical universe became physical, as opposed to creation event, you can find the TOE begins to develop nicely. Infinity is a friend of our quest for truth.

  • Frank_Truth

    Infinity and infinitesimal could exist but they have never actually been observed. I would use Occam’s razor to be skeptical of the existence of their existence. That being the case, I think it is like that every possibility is a reality. I believe Hugh Everett’s MWI is correct, along with Hartle & Hawking”s NBP, and all Max Tegmark’s universes.

  • GorgonPolus577

    If the universe is not infinite in time and space then what do we have at the end of time and space? Anything that ends in time has something that comes after the end. The end of space requires a boundary with something on the other side. The universe is not the universe without infinite space and time.

  • davidt

    Pythagorean nonsense…..

  • DJ

    Give the quantum nature of reality, infinities fit perfectly – they simply express the uncertainty of what we, laughingly, call local reality. Simply put, there is no way to localize a mathematical point.

  • John Dodero

    “You can’t solve a problem from within the context it’s created in” A.E

  • johnnysy

    Infinite universe doesn’t exist it is finite. Easy way to explain. When matter expands beyond its capability to retain energy then is finite. All matter decays and loses it energy. In time because of the drifting apart of galaxies there will be no energy to maintain a structure for matter.

    This will affect the stars, planets, and in the end black holes. If all matter doesn’t exist in its present form then there is no reason for the universe to expand thus it becomes finite. Nature shows this to us on a daily bases when things live and die. Sorry if you were planning on thinking about being immortal.

    I could go on and in simple terms explain a lot more but it will either get you pissed off or very depressed when it shouldn’t do either. Everything lives, everything dies, everything starts and everything starts. The only thing that is infinite that we hope on that this cycle will continue. One universe ends another begins.

    The Universe is a dog trying to catch its tail. It will never catch it but will become tired then eventually stop trying. Then it will die. Energy will no longer be sustained. Have a nice day.

  • ronaldmsonntag

    Can anyone explain to me why Heisenberg’s uncertainty principle, which blows up momentum at a point, doesn’t stop the theoretical singularity of a black hole from forming? That is a classic examples of two supposed infinities in direct opposition.

    • YeahRight

      It does. There is no singularity in any conceivable quantum mechanical version of a theory of gravity. Strictly speaking, there are no singularities in general relativity, either. The one and only way to get a singularity is to pick a very special parameter set (total charge and angular momentum have to be exactly zero), which is simply not the case for any realistic physical system. This leaves the singularity as a trivial mathematical artifact.

      • ronaldmsonntag

        Then I wonder why so in so many discussions about black holes, even from notable scientists, they almost always describe matter as being crushed into a singularity? Why give the general public such a terribly wrong model?

        • YeahRight

          For one thing, violence sells. If I told you that for a physicist all states of matter/spacetime are equal and that we actually see the universe as a continuous change of scales (divided into “eras” defined by the predominant physical effects driving the dynamic) which only appear to be “violent” in the beginning because humans have an observer bias for one specific set of scales, would you buy my book? I don’t think you would. You want to hear fairytales, so the colleagues of mine who make a living with writing layman books will tell you some. Do I find this a good idea? No. Am I going to change the publishing business? No. Can you get the right answers from me if you ask? Yes.

  • Anton Szautner

    ‘Infinity’ isn’t required to ‘exist’ any more than ‘nothing’ is required to. But that does not mean that these mathematical concepts are not useful and don’t supply a meaningful context for the ‘existence’ of the definable or tangibly finite. If nothing else, in general, if infinity shows up in a physical theory, its telling us something important is missing, or that we don’t understand it.

  • Jim

    Infinite seems to me a possibility that nothing can ever achieve even in the most dramatic of circumstances. It has to be possible but also impossible to reach. Nature can therefore act according to the laws and at the same time act within a boundary.
    The possibility of infinity exist but the achievement of it is impossible.

  • Randall Wakefield

    “The past, like the future, is indefinite and exists only as a spectrum of possibilities.”

    Stephen Hawking

  • Randall Wakefield

    “Only two things are infinite, the universe and human stupidity, and I’m not sure about the former.”

    Albert Einstein

  • Jacobo

    We really need to stop treating infinity as an object. Infinity is a process. In mathematics infinity only has meaning when it is accompanied by a limit. For example limit x->oo of f(x) =L means that for all epsilon there exists delta such that for all x>delta abs(f(x)-L) < epsilon. This is a process where you can keep on picking a smaller and smaller epsilon and when the limit exists you will always find a delta where abs(f(x)-L) < epsilon. This is analogous to what John is trying to say. At some point we are doing physics with such a small epsilon that we are not able to find a delta, and the process of going to infinity fails. It is not that some value or object that is infinity does not exist, rather our mathematical procedure fails.

  • Joshua Kellerman

    idk, circles and pi seems pretty useful in describing a photon sphere, and that’s an infinite number, somewhere in which is my DNA sequence.

    • YeahRight

      1 is a definition, so is pi.

      • Joshua Kellerman

        This article is writing to say that “the universe is rational. There is a smallest unit of space.”

        If there is a smallest unit of space, you can never approximate a circle perfectly. It will always be off a little tiny bit. 3.14159 might be good enough for practical applications, but after billions of years, a particle traveling on that circle will gain error – (because it couldn’t actuate pi perfectly when it was off just by one plank length, because pi needs an infinitely small unit of measurement to be defined) – just like compound interest and spiral out or in. In a finite universe, there is no such thing as a perfect circle. If there is no such thing as a perfect circle, where do we even get the concept from?

        Also, no, pi is not a definition. If I asked you to define pi, you would die before you got too far. It is, and will always be, an approximation. 1 is a definition.

        • YeahRight

          The article is full of it and the author simply misrepresents the ontology of numbers in physics. All he proves is that he is clueless about how physicists work with theories and models.

          A circle is a mathematical object. There never were any circles in nature and physics has never pretended that there were any. Physics simply says that under the right circumstances a sufficiently round physical object can be modeled with a circle and that that model will give sufficiently precise results. The word “sufficiently” in this case means that someone has done an experiment that has confirmed that a circle is a good enough description of some roundish thing.

          That’s it. Anybody who doesn’t get that doesn’t know physics.

  • Ken Fritz

    The concept of infinity may not compliment our approach to physics but that in itself is not a reason to dismiss it’s existence.

  • Artur Sixto

    The idea of infinity, I see it neither as an assumption about the natural world, nor as a mathematical aberration.

    Regarding the natural world, most of times the assumption has been its finiteness, even in absence of apparent limits. The Earth for instance, was often speculated to be flat with edges over which one could fall. The skies were conceived as seven concentrical spheres beyond which there was nothing, and it would be tricky to dispute that nothing being infinite fails to pose an infinitude issue. Until only decades ago the default assumption about the natural world was its finiteness. It was born from a singularity 13 billion years ago – nevermind singularities are mathematical, physically unproven. It had an age and some expansion rate and could therefore never be infinite. We could calculate its limits, even beyond the observable universe. It was just a new version of the flat Earth with edges, only this time with a mathematical model attached. Of course we were all left absolutely free to wonder if that was really it, why some expanse from ours there should not be other singularities and universes. We could speculate about infinitely many cycles of universe expansion and collapse. We did speculate about it, as we did develop mathematical approaches that allowed for infinitely many universes to coexist in parallel or in succession with different laws of physics or not, giving rise to all, infinitely many, possible outcomes.

    As for mathematical infinity, if we are able at all to conceive abstract infinity (in terms of say, space or time) irrespective of mathematics, or else in terms of numbers, I cannot understand how infinity can ever be regarded as mathematically aberrant. It might leave us perplex or helpless when we come across it in maths, but it sounds arbitrary and just too convenient to reject it. And how would you reject it anyway? Shall we reject zero as well?

    Regarding the infinitely small and the infinitely big, I am surprised to read that physicists do consider the natural world’s potential for infinite

  • modmans2ndcoming

    So, how much energy is needed to accelerate a mass to the speed of light? GR says an infinite amount of energy. Discarding infinity as a concept in nature will require a lot of reworking of fundamental theories.

    • YeahRight

      OK, so you don’t know how to think logically. Relativity says that objects with rest mass can not be accelerated to the speed of light. There is no infinity anywhere to be seen in that statement.

      • DarkMatterExplainer

        Actually there is, the speed of light is the limit because it would take infinite energy to get there, therefore it’s impossible for anything with any mass. That doesn’t need to be reworked at all.

        • YeahRight

          You too are having difficulty with physics, Brutus?

        • albertobladerunner

          You don’t know that, you just want to think you know it.

      • albertobladerunner

        Relativity is just an approximation to reality, not reality.

    • albertobladerunner

      Who said the “fundamental theories” are religious commandments? in science what is fundamental is that anything can be questioned.

  • Duke Eyman

    It will always amaze me that man believes he has the ability to know the unknowable.

    • YeahRight

      It doesn’t amaze me that you didn’t take the time to study science.

    • albertobladerunner

      How do you know there is such thing as the “unknowable”? Just because something is not known in your lifetime that does not prove something is “unknowable.”

  • ThatCat

    The definition of infinity at its root means unfinished or incomplete. And now it means to go on without end or limitless or immense. That’s stupid. It doesn’t mean limitless (no limit) or immense (big). How to you turn in = not and finite = finished into limitless and immense? It’s no wonder they’re having problems. Doesn’t take rocket scientist.

    • YeahRight

      Who is they? :-)

  • Enézio E. De Almeida Filho

    Who sez infinity is ruining Physics? Theoreticians are ruining it!!!

  • Ken Weiss

    In itself, this is a bit misleading. Infinity is treated as being a limit, not a ‘thing’. This is similar to ‘imaginary’ numbers. What I’ve heard physicists say is that in the center of a black hole or at the very beginning of the Big Bang, we cannot say what reality is, because these are infinite points mathematically, but unknown physically. Equations, as in quantum mechanics etc., let us deal with things we cannot really intuit. Inflation says new ‘space’ emerges out of absolutely nothing, which is another element that we cannot understand by intuiting.

    Wolfram argued that existence was pulsed, in discrete units rather than infinitesimals, but then what is in between the units? These things seem more like quantum mechanics than actual errors: things we can’t (yet, at least) ‘understand’ the way we understand things we can touch and feel in the usual way.

    So, things are strange but we need to be wary of being too wary.

  • God’swill Doctore Abang

    Infinity is only a name for an idea that the human mind can’t comprehend, but it is there , Undeniable.

    • YeahRight

      Infinity is actually very well comprehended… by people who went a little further in life than failing high school.

      • God’swill Doctore Abang

        is it really? Or just a bunch of equations and ideologies which I do not undermine, my point is no matter how much we study and practicalize there is always a limit, hence; Infinity.

        • YeahRight

          Like another poster remarked above, that was beautiful hogwash.

  • GarretKadeDupre

    Max, you aren’t making any sense. You say Planck confirmed the Big Bang, but you know FULL WELL that Planck falsified the Big Bang’s prediction of large-scale uniformity with those darn multi-poles, which you are on record acknowledging in The Principle Movie.

  • Lawrence Cooper

    Infinity – just a way to express something so immensely large or small that you can’t quite grasp it. See. Simple.

  • Tony Smith

    Wow, two huge misconceptions in the one article correctly debunking any physical realisation of the mathematically useful concept of infinity.

    Citing Cantor but missing the point entirely that the uncountable number of possible arrangements grows incomparably faster that the countable number of actualised arrangements leads into the popular trap of imagining macro scale objects can be independently duplicated.

    Suggesting the fabric of “empty” space is being “stretched” without any hint of evidence that the array of measurable properties of that space are proportionately different to what they were when our post-inflationary bubble was less than 10% of its current age. Space being produced, not stretched, drives the expansion. This is one of the easier things to model with discrete micro scale processes.


    What they are saying is? They can’t measure it today! But what about one year or ten years or maybe 100 years? Are they saying that we know all the information that is or will be available forever? It seems to me that the statement is kind of ??? Nobody knows what will be found out in the future? Like maybe light travels a a different speed moving away from us? or some other weird action the we don’t know about at this time.

  • ericlipps

    Saying “Infinity doesn’t exist” doesn’t make it go away. Neither does the fact that we can get useful answers to many questions by merely dealing with pseudo-infinities, that is, very large (or very small) numbers. Those answers are approximations, and pretending that they aren’t or that it doesn’t matter that they are is likely to have a price.

    Division by zero is another matter. Consider the following:


    If dividing by zero is considered to yield infinity, then you do run into problems. Take the above “equality” and multiply bvy zero:


    Now subtract 1 from each side:


    What you’ve “demonstrated” is that every number is equal to every other number (since the difference between any two numbers is equivalent to zero)–in which case mathematics disintegrates.
    Infinity has to be considered as something different: a limit to be approached but never quite reached. That doesn’t make it any less real in philosophical terms, or even in mathematical ones.

    • YeahRight

      I don’t think that’s correct. Infinity can be added to the set of the complex numbers in a process called compactification. If I understand it correctly, not every topological space can be made into a compact space by compactification, so the question whether “infinity” exists as an indivdual element of a space is a non-trivial one that has deep consequences for the structure of the space. Having said that, IANAM (I am not a mathematician), so you will have to read up on the details yourself.

  • David Kline

    Perhaps Mr. Brockman will move on to question the square root of negative one.

  • j2saret

    Next we’ll hear that quantum is not spooky. Its just inadequate measuring instruments. Gods dice could some day be seen.

    • YeahRight

      Quantum mechanics isn’t spooky. The only thing that is spooky is how hard it is to explain to most people.

      • waffleironmarch

        Spookiness is by definition a human mental state response to the way a counterintuitive concept affects our psychological makeup, so if someone says “that’s spooky” it’s equivalent to saying something like “that’s funny,” which is generally taken by non-autists to be an indicator of a personal emotional reaction to a stimulus with implicit underpinnings of subjectivity, rather than a universally prescriptive, objective assessment.

  • daqu

    This is astonishingly ill-considered hogwash.

    * “Carl Friedrich Gauss denied that anything infinite really existed”:

    The fact that venerable mathematicians of the past did not believe in infinity has no relevance to whether the concept is valid: in every subject, many venerable scientists of the past did not believe in what modern science has discovered.

    * “we have no direct observational evidence for either the infinitely big or the infinitely small”:

    Well, duh. We have no direct observational evidence of anything that we can’t measure. But this speaks only to human beings’ ability to measure certain things, not whether they exist.

    * “Without the infinitely small, inflation can’t make the infinitely big, so you get rid of both infinities in one fell swoop—together with many other problems plaguing modern physics, such as infinitely dense black hole singularities and infinities popping up when we try to quantize gravity.”

    More nonsense. Declaring by fiat that infinity does not exist will NOT make it disappear from those equations. No one knows what those infinities mean, but they still have to be explained.

    * “air of course isn’t truly continuous. I think it’s the same way for space, time and all the other building blocks of our physical word.”:

    No disagreement that the properties of air are a consequence of its composition as a collection of molecules. But this is no argument for whether or not space and time are continuous rather than discrete. Tegmark’s saying he “think[s] it’s the same way” is fine, but that does not contain any reasoned argument.

    Tegmark is showing once more that he really should stick to his specialty rather than to engage in poorly thought out philosophical speculation.

    Infinity in mathematics is a supremely established concept regarding the size of sets. Even infinitely large and infinitely small numbers are on a totally solid logical foundation.

    There is no current reason to assume that the universe could not be infinitely large.

    (As usual, Tegmark misstates the implications of an infinitely large universe:

    * “inflation predicts that there will be infinitely many copies of you far away in our infinite space, obtaining each physically possible outcome”.

    Not at all. What is known is that — with probability = 1 — whatever happens in an infinitely large universe, somewhere else there will be events arbitrarily close to those.

    * “If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information”

    OK, this is true. But why does there need to be any description of that distance?

  • boonteetan

    Infinity, a very beautiful mathematical concept, is one of the fundamental pillars of math. Pi and e are two most important infinite numbers. They are essential in physics equations. Can physicists do away with infinity? Don’t jump into conclusion too fast too soon.

    • YeahRight

      Neither pi nor e can be found in nature. The only places where they appear is in simplified mathematical models of nature.

  • Barry Rudd

    Look at it this way. A 2 dimensional being on s sperical surface, that has a finite surface area, can move around and never reach an edge.
    Now consider a 3 dimensional being (us) moving around on/in a finite volume of space will also never reach an edge.
    The analogy is: 2 dimensional spherical surface to a 3 dimensional spatial volume. Both finite and both never ending.

  • Randall Lusk

    And GOD said I am the biggining

    • YeahRight

      He forgot to tell you to use a spell checker.

      • Randall Lusk

        When you are righting for morons it does not really matter. Does it?

  • Randall Lusk

    And GOD said I am the beginning and the end, the first and the last. Infinity ad infinitum.

    • YeahRight

      Thor said that?

      • waffleironmarch

        You think you’re so edgy.

  • László Tóth

    Our system has a frame.

    Just inspect it from the physical zero state which is a Space(time), Energy(matter) Information free nothing.

    If we are able to assume when and how the origin happened, with the information about the exponential expansion of space and the linear information about it provided by time we can suggest the size of the universe.

    The truth is that space is ever expanding. It is existing as a “separate dimension” from materialised informative energy. In other worls space will expand even energy collapses to matter.

    Time will never stop since something will always exist. But it does not mean that it is infinite. It has an exact value/information about the sytem. It started at one point and will continue for ever but it is a precise information every time we inspect it. We are just not yet able to make sense and calculate it´s real information.

    So the Universe has an ever changing but existing frame. Provided by the information about the starting point the physical zero state and the information about the rate of expansion of low energy and matter free space plus the information about how long the system has existed.

    For the right explanations we will have assume the first physical steps of the delicate universe which explanation should suggest the origing of intelligence, space, energy, matter, gravity……

    For the right explanations we have to refine some of the current mathematical understandings because the first physical process discribed by a mathematical operation will look like this



  • EquusMtn

    Ok, I’m no mathematician, but doesn’t any number, variable, or quantity divided by itself = one (1)?

  • Michael Duke

    Inifinity is easy to understand; add another zero or decimal place. No infinity places ‘Limits’ on everything and places a description on assumptions. If there is an end to space or quantum theory it’s in your imagination.

  • himanshu khatri

    by basic concepts in x/3 we make 3 same parts of x but in 0/0 how can we make 0 parts of 0
    i mean if there’s nothing to divide how will you divide it in any parts so 0/0=0

  • Thomas Foster

    Space cannot be infinite, and neither can time. Infinity is a metaphysical concept that is beyond space and time. Both space and time imply a limit essentially, whereas infinity as a concept means limitlessness essentially. Therefore space and time are irreconcilable with the concept of infinity. If infinity is, it must be beyond space and time, i.e, it must be spacelessness and timelessness. An indefinite extension of time would be indefinitude, not infinity, because time had a beginning, or it has proceeded from a first instant. True infinity would not proceed from a first instant, or be a succession of instants at all. It would have always been timeless. Similarly, an indefinite expansion of space would be immensity, not true infinity, because space had a beginning, or it began expanding from a point. True infinity would not have expanded from a point, or be expanding at all. It would always have been limitless. This is a mistake people often make about the metaphysical concept of infinity. It can only be a metaphysical concept because on principle it is beyond rational physicality, which is why it was always associated with theological ideas of God and metaphysics in general.

    • waffleironmarch

      But something that is infinite in any direction, regardless of its point of origin is, by definition, infinite. Infinity divided is still infinity. The whole “indefinitude” thing is pure semantics.

  • Muhammad Shatry

    Infinity started with the Big Bang and ends with Black Hole with Everlasting Repetitive Phenomenon.

  • j3relowf

    How are they going to explain the improbable fine-tuning of our universe to support life if they don’t have infinitely many universes to make ours probable?

  • daqu

    The main argument Tegmark uses here is patently ridiculous: That because the current state of physics can’t make calculations about a certain phenomenon, therefore the phenomenon probably doesn’t exist.

    What hogwash! The universe, I am quite sure, was not designed by nature with the ability of physicists to predict everything about it as one of the design goals.

  • Douglas Broccone

    Not capable of higher math myself, but it seems intuitively, that there must be something to the notion of an infinite universe. the Author here is making a practical argument…If the concept of Infinity doesn’t work to easily predict things here at our local piece of space, then lets scrap it and pretend it’s like what we seen in front of us… rubber band, q tip, whatever… that’s sensible, for the purposes of getting a finite answer… but , if you do a mental experiment, just going in any direction … eventually you assume you will reach the end of something, the land, the ocean, the atmosphere… since it’s not practical to continue farther, you stop, and there’s the end of the Universe? So we send probes, we launch a telescope, we observe / calculate radiation, ok… that’s the end of it… yet we keep learning and creating more sensitive tools, we seem to extend the edge of the universe the smaller and the larger, pretty frequently…
    I would bet the opposite of what the Author contends… I think that if we could go in all directions and scales, simultaneously, forward and backward in time… with impunity… disregarding our physical bodies… there would be no final edge, only more … and less of everything… the same things and the opposites of all things thought of and not… completely everything… I just can’t figure out why it isn’t clear already that all things are part of a continuum… All I can think is , it must be so… if you can point to a direction, there must always be something beyond… if you can conceive of time, there must always be now and later… and past must have always been… It couldn’t have just begun from nothing… There is no “Nothing”

  • Franck Merlot

    At least equally meaningless as infinity is the concept of non-existence as in “zero of anything”. Since it can never have a reference point, non-existence is not anywhere, hence nowhere. If it can’t be anywhere, it can’t exist.

    As reductionist as it might seem, it is actually logical. That would finally put the tired old questions about “creation” to rest.

    This is a wink to Krauss and his A Universe from Nothing and especially to the creationist nutjobs.

  • Paul Bristol

    I’m startled to see Tegmark claim that physicists think the mathematical infinite exists. On the contrary; the mathematical infinity is an abstraction. Logically sound and endlessly fascinating; but no physical existence whatsoever is claimed for it. Physics uses math to model its experiments. Mathematicians don’t confuse the map for the territory. Tegmark, believing that physics “is” math, does think the map is the territory.

  • disqus_zXLbNfw1Yi

    It’s turtles all the way down.

  • Rocky




  • Franck Merlot

    Not all numbers are concepts. Everything in the universe can be described in mathematical terms or in any other abstraction used by any other intelligence in the universe describing the same physical reality. Such abstractions, whether they might be expressed in human maths with numbers and formulae or any other expression which describes the same reality can be matched up.

    If you describe a planet around a solar system with a “bar” (-) then any intelligence would be able to decipher that – – – – – – – – means “8 planets” in human terms.

    Now try that with infinity or zero in terms of non-existence.

  • the truth

    The author is wrong. He or she is overlooking one extremely important point. The fact that it is not possible to measure a single point. Let me explain, the very act of measuring requires two points. You must measure from point A to point B. Therefore, one is infinite by definition. It is immeasurable, and inconceivable. In reality only one exists. Every other number is no more than a division of the original whole. For instance, instead of 2 its 12, not 3 but 13. This is what physics needs to move further. Oh and by the way, the challenge to physics is to find the unifying equation of all equations. The author needs more work.

  • the truth

    Infinite is describing only one thing and that is the number 1. The number 1 is the largest of all numbers. The numbe 1 is infinite. By definition infinity is immeasurable, and inconceivable, and goes on for ever. Well this describes only one number and that is the number 1. The very act of measuring requires two points. You must measure from point A to point B. Regardless how small the distance is. Therfore , 1 is infinite and all other numbers are no more than a division of the original whole.

    • diogenes

      Ancient Monism. Measuring is an empirical act attempting to create an abstraction that approximates what others who share the physical act corroborate. The map is not the territory.

      “When I use a word,” Humpty Dumpty said in a rather scornful tone, “it means just what I choose it to mean — neither more nor less.”

      Disagreement, as to what “infinity” “is” ignores the consensus nature of language.

  • bt

    Surreal numbers solve this problem. The problem with infinity/infinity or 0/0 is that these aren’t accurate. Students are taught that 2 times infinity equals infinity – but that’s pure nonsense. Infinities and infintesimals are only meaningful when cancelled out by other infinities and infintesimals to give you a real result.

    In otherwords, we as a society use “infinite” to really mean “really really big”. And so obviously if you multiply something really big by a constant, it remains “really really big”. Its true, just not at all accurate.

  • Jim

    The edge of the universe is like the end of this comments section. As long as we keep trying to find it never comes. It’s only there when we stop looking.

    • Wizardofmann

      I can wholeheartedly agree with the second sentence but shake my head vigorously at the other two. It matters not how long you try, how clever or intelligent a scientist you may be, humans will never find the edge of ‘space,’ as it simply does not exist. The answer may not be palatable but the answer is simply that humans are not intelligent enough (And never will be) to digest what ‘infinity’ actually is. Infinity, time, continuum etc etc are ‘invented’ by humans. As the ‘good lord’ apparently said, ‘I have no beginning or end’ (Or words to that effect) Could this be the end of the comments section? Nah, ‘human instinct’ says otherwise.

  • Ravi Sadana

    If anything needs retiring, it is the phoney “Time”, the socalled incremental dx/dt function. If we take “time” out of everything, we’ll attain true relativity.
    The self-contained and self-serving edifice we have built since the days of Newton needs a thorough review. We scientists have failed mankind by advocating blatant materialism. Human beings have become slaves of “time”, “Money” and “mathematics”, and, becoming dumber year by year.

    As proof, I offer the accomplishments of prescience civilizations. See what they achieved with a copper chisel.

    When I read some comments, I see children fighting over marbles. This elitism, this esoteric knowledge called mathematics is not serving mankind. It’s self-aggrandizement. It’s egotism of the worse kind.

    What we have done over the last 250 years is interference with the natural evolutionary processes by insidiously conditioning the human brain-mind complex by overfocussing on “time”, “money” and “mathematics”. Just look around. Everyone is a slave of “time” and “money”. Everyone considers “time” as the sole undercurrent of universal change.
    “Time will heal” “it’s a question of time” etc. etc.
    Nothing of the sort. Atomic and molecular interactions, totally free of socalled “time”, cause mutations. Processes of biology and physiology heal us or make us sick. “Time” has nothing to do with anything in nature. It’s a bogus concept, invented by science to propagate its causes.
    All measuring methods of time are based on motion.
    Do you know that even before space came into being, “Dynamika” an entity by itself, prototype of “Motion” came into being first?
    We are wrongly focussed on mass moving through space. Because it is easy to measure.
    What’s hard is to measure is movement of space. Space is the medium of expression of matter. All matter came from space. Everything exists in space and space exists in everything.
    The difference is “captured space” and ambient space “coursing” through the empty corridors of matter.
    These concepts are being ignored as we have it easy with “time”.
    Read my paper “The Living Grand Unified Field Theory”.

  • Recidivist

    It is certainly true that Physics does not require infinity. But is the author actually suggesting that there is a largest integer? Or simply that we dispense with mathematics?

  • antonio carlos motta

    I think that the infinity exist to explain the asymmetry of the physical structures,as well as the infinities shapes exist in the universe.the complex numbers are linked to the fourth dimensions einsteinnian for the left handed into right handed and vice versa ,does appear the rotational invariance non totally symmetric so,where appear the chirality,and in the breakdown of symmetry as cp and pt could appear the particles geometry and topology,so as the multi universe generating the super symmetry to explain the conservation of the rotational invariance,so as the spins,or why the rotational invariance need to be broken.Then the PT symmetry breaking makes connect the space and time for spacetime continuos,doing appear the Constance of speed of light due at the breakdown Of PT and the asymmetry among particles and antiparticles,through the operator CP.without the infinity is impossible the physics create the several shapes in the universe.the infinity does exist such the infinitely largest as infinitely smallest.

  • antonio carlos motta

    If 0,0009…is equal 1, then -0,0009…is equal – 1,sure?

  • ilduderino

    Or maybe only infinity exists and you can only quantify things in isolation but not in totality and only in approximation and not with precision. Science demands linear reductionistic deterministic mechanical quantification and there’s an aspect of reality it does not fit this kind of paradigm. Maybe science should ignore Infinity at least for a while but when the doors of perception are cleansed there it’ll be in a grain of sand and all that…

  • Elelei Guhring

    A Mathematician, Physicist and Alcoholic walk into a bar…

  • DB

    Sound like Physicians are BUUTTTT hurt about an idea. When one hears “This Idea Must Die”, one knows they are as regressive as a pope in the 9th century.
    How about this idea? Make your own idea and disprove the idea of infinity with evidence and let every generation disprove you with evidence.

  • antonio carlos motta

    if the real Numbers or continuum are greatest of the infinity,same that can there are others infinities for beyond the continuum.But there is one dimension,two dimension and infinities dimensions,between the dimension there is a discreteness between theirs,and in These discreteness generate the points-holes are possible complex Numbers ,with the component the coordinate imaginary or the time,that are reflected in the transcendental points connecting the left handed and right handed systems,there show that shapes could be modified by deformations smoothly in the transictions between the different shapes between the points -points.the non linear álgebra topology geometry are showing the connections between the discreteness and possible that exist the time as dimensions splitted and generating simultaneously the non orientable and orientable space as motions curving the space pseudo flat. No could divide infinity by infinity,because infinity is just a word that implies the loopholes to explain that the infinities has different “Potentia”

  • antonio carlos motta

    The thought in Physics is an ideal thought, while the mathematical thought is real, it is the Infinity is the thought deeper of the creation of that would occur in the universe, the infinity is one but is the zero when think in mathematics

  • Connor Prewitt

    We must be in the Matrix and this post was meant to take us away from the allure of finding the truth. Perhaps simple physics are the way our controllers want us the think, whilst infinite is the answer to how we get free!… I watch too many movies

  • David_Rogers_Hunt

    I am intrigued by the idea that if Infinity doesn’t actually exist, then Godel’s Essential Incompleteness Theorem doesn’t apply to physicists’ “Theory of Everything” because all the paradoxes that drive GEIT derive from acknowledging and accommodating the infinite set of integers as an actual real thing.

    • David_Rogers_Hunt

      ……………………..a = b
      …………………a * a = a * b
      ……….a * a – b * b = a * b – b * b
      ( a + b ) * ( a – b ) = b * ( a – b )
      ………………..a + b = b
      ………………..b + b = b
      …………………2 * b = b
      ……………………..2 = 1 ?!!?

      x / 0 = infinity,… really?

  • Wizardofmann

    I wish I could find the quote on here where someone wrote: ‘Someday we (I have to presume that means us humans) will do something really stupid one day.

    ‘Whoever wrote the above quote, has obviously never heard of Hiroshima and Nagasaki, the Cuba crisis etc etc etc!

    We are all stupid and always will be, forever, or until Nature decides She has had enough of our stupidity and wipes us from the face of the earth. Human nature dictates that man is born stupid and always will be. The is no mathematical equation/calculus to prove or disprove, nor will there ever be, which will reveal infinity (A man made word, as is time and every other word in out dictionaries) simply because it does not exist! Put it this way, for ever means for ever and ever and ever. And ever end ever and ever…………


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