Tuesday, October 2, 2012

Reality: Is everything made of numbers?

Amanda Gefter writes in the current NewScientist:
How is it possible that mathematics "knows" about Higgs particles or any other feature of physical reality? "Maybe it's because math is reality," says physicist Brian Greene of Columbia University, New York. Perhaps if we dig deep enough, we would find that physical objects like tables and chairs are ultimately not made of particles or strings, but of numbers.

"These are very difficult issues," says philosopher of science James Ladyman of the University of Bristol, UK, "but it might be less misleading to say that the universe is made of maths than to say it is made of matter."

Difficult indeed. What does it mean to say that the universe is "made of mathematics"? An obvious starting point is to ask what mathematics is made of. The late physicist John Wheeler said that the "basis of all mathematics is 0 = 0". All mathematical structures can be derived from something called "the empty set", the set that contains no elements. Say this set corresponds to zero; you can then define the number 1 as the set that contains only the empty set, 2 as the set containing the sets corresponding to 0 and 1, and so on. Keep nesting the nothingness like invisible Russian dolls and eventually all of mathematics appears. Mathematician Ian Stewart of the University of Warwick, UK, calls this "the dreadful secret of mathematics: it's all based on nothing" (New Scientist, 19 November 2011, p 44). Reality may come down to mathematics, but mathematics comes down to nothing at all.

That may be the ultimate clue to existence - after all, a universe made of nothing doesn't require an explanation. Indeed, mathematical structures don't seem to require a physical origin at all. "A dodecahedron was never created," says Max Tegmark of the Massachusetts Institute of Technology. "To be created, something first has to not exist in space or time and then exist." A dodecahedron doesn't exist in space or time at all, he says - it exists independently of them. "Space and time themselves are contained within larger mathematical structures," he adds. These structures just exist; they can't be created or destroyed.

That raises a big question: why is the universe only made of some of the available mathematics? "There's a lot of math out there," Greene says. "Today only a tiny sliver of it has a realisation in the physical world. Pull any math book off the shelf and most of the equations in it don't correspond to any physical object or physical process."

It is true that seemingly arcane and unphysical mathematics does, sometimes, turn out to correspond to the real world. Imaginary numbers, for instance, were once considered totally deserving of their name, but are now used to describe the behaviour of elementary particles; non-Euclidean geometry eventually showed up as gravity. Even so, these phenomena represent a tiny slice of all the mathematics out there.

Not so fast, says Tegmark. "I believe that physical existence and mathematical existence are the same, so any structure that exists mathematically is also real," he says.

So what about the mathematics our universe doesn't use? "Other mathematical structures correspond to other universes," Tegmark says. He calls this the "level 4 multiverse", and it is far stranger than the multiverses that cosmologists often discuss. Their common-or-garden multiverses are governed by the same basic mathematical rules as our universe, but Tegmark's level 4 multiverse operates with completely different mathematics.

All of this sounds bizarre, but the hypothesis that physical reality is fundamentally mathematical has passed every test. "If physics hits a roadblock at which point it turns out that it's impossible to proceed, we might find that nature can't be captured mathematically," Tegmark says. "But it's really remarkable that that hasn't happened. Galileo said that the book of nature was written in the language of mathematics - and that was 400 years ago."

If reality isn't, at bottom, mathematics, what is it? "Maybe someday we'll encounter an alien civilisation and we'll show them what we've discovered about the universe," Greene says. "They'll say, 'Ah, math. We tried that. It only takes you so far. Here's the real thing.' What would that be? It's hard to imagine. Our understanding of fundamental reality is at an early stage."
My FQXi essay argues that reality, at bottom is not mathematics. I say that Tegmark is wrong in the most extreme way. There is no real object that is also a mathematical structure. Not even an electron or a photon.

This is the last week for the public rating of the FQXi essays. My essay has consistently been in the top 10 on the community rating.

Tegmark says, "we might find that nature can't be captured mathematically," but I say that has already happened with quantum mechanics. Niels Bohr said:
It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we say about Nature.

Everything we call real is made of things that cannot be regarded as real.
I think that Bohr was saying that observations about nature can be described mathematically, but nature itself cannot be captured mathematically.


  1. Surely mathematics is similar to language.

    It can be used in limited ways to describe physical reality or to tell stories of imaginary and unphysical realities. As is the case with language.

    There are however, many aspects of reality it can't describe faithfully and nothing it can describe completely. As with language.

  2. I have been bumping into this delusion for quite some time in my reading into physics and philosophy, that somehow labeling or measuring something is reality. I don't believe there is evidence the physical universe works like the Looney Toones cartoons where people run off the sides of sheer cliffs and continue to run normally until someone jokingly motions for them to look downwards.. then they hang for a moment as reality sets in before falling. Any blind man will tell you, reality does not wait for you to catch up before you take a fall. Likewise, taking measurements of something might define it spatially within some meter, but it does not change the object measured, and calling manure a rose does not make it smell like a rose. Now we have physicists and mathematicians of all people telling us one dimensional strings can have the tensil strength greater than any known substance... what rubbish. One dimensional is like a line, a length, an inch, a millimeter, a mile. How much tensil strength is there in a mile? Can you hang or suspend from a length? Only in cartoons...and modern physics apparently.
    The belief that math is reality is nothing more than a sad variation of Platonism, anyone who knows their philosophy would see this and avoid it like the logical sinkhole it is, as it leads nowhere and precludes the possibility of understanding. Unfortunately, many of the highly educated of today seem to hold philosophy in disdain, and so they are unprepared to recognize an old idea wrapped up in the new veneer of maths.

  3. Sometimes the truth is just "Bohring". The narcissist hates any kind of limit to our understanding, because he wants to return to an infantile feeling of omnipotence. I think psychology has a great deal to tell us about the regressive psychosis we are witnessing, even in the hard sciences. Our culture embraces what John David Ebert calls "the cult of the wonder child" and we rarely recognize how much we owe to the past. Christopher Lasch was way ahead of this declining culture.

  4. Tegmark's "conclusion" is not the only one. One might say that there is ONE mathematical object that defines all the particles, and the REST of math is basically junk that has nothing to do with what exists in the physical world. I do not see any justification for saying that the rest of math is in other universes. It sounds like he is trying to justify not only the string landscape but any other notion that might make 'mathematical sense' - like replacing real coordinates with finite fields to make more landscapes.
    Describing is not the same as defining. Physics is mostly about describing - Einstein describes the expansion of space, but does not explain how this occurs. The problem here is that any look at the standard model reveals that it is wedded to very fundamental algebra. The SM is not committed to 3 generations of fermions, but it probably should be considered incomplete unless it can specify why there are 3 generations. Whatever the reason, one expects it to make algebraic sense, not some vague 'mathematical' sense, like 3 being a prime number.
    The notion of objects 'made of numbers' is goofy. Algebra is primarily about structure - like Clifford Algebra and the structure of space. What Tegmark seems to be saying is that ALL Clifford algebras have some kind of reality - but what does that mean ? Is it to be taken as saying that n-space is 'real' as if Nature can not make up its mind how many dimensions there are ? Of course, a different interpretation is that Clif is N-body instead of N-dimensional - as in atoms with N electrons - as in the Brauer-Weyl paper 'spinors in n-dimensions'. So, a lot of the problem here is interpretation. Often mathematicians do not know or care, but to a physicist one does care whether the issue is n-electrons or n-dimensions.
    Wheeler's 0=0 is just plain nonsense. What the heck has that got to do with complex numbers and quaternions, or rotations in three dimensional space ?
    The idea that mathematical structures 'just exist' and can not be created or destroyed is also a bit weird. You could say that a relationship 'just exists' - but you can turn it on or off. You could say that a spin-up and spin-down electrons are actually different beasts that differ only in spin, or you could say it is a 'degree of freedom' and there is just one beast.
    What bothers me is that physics does not even take Octonion algebra seriously, and that is elementary stuff. Oh, there have been various ideas - like octonion Hilbert space, which seem to go nowhere. Heck, a lot of physicists thought quaternions a "hermaphrodite monstrosity" , but now quaternions are all over the place in physics - though not in the way Hamilton was doing things. It took decades to make reasonable sense of it. That seems to be an object lesson.

  5. You math-heads are all overcomplicating my universe ("my" as in "the one I inhabit", not "my" in the sense of creation and/or ownership, heh!). Ayn Rand said it best: "A is A." Posed: "What is reality?" Answer: "Reality is Reality." Which is to say that using any memes, symbols, strings, etc. other than "reality" in efforts to define "reality" is merely analogizing and/or instantiating; a synecdoche BTW being both analogy and instantiation. Still, it's fun to kick words around in efforts to escape from what I term "The Tower of Babble." Seems inescapable; describing is never defining formally. These days I no longer argue with anyone about anything, I just offer operational definitions and go back to work.

  6. These physicists are hopelessly naive philosophically. This "desperate platonist" stance seems to be a perennial attraction for scientists and mathematicians, but it is simply a confusion of the necessarily logical way we structure our world. It's not the least bit miraculous how physical theories accord with maths time and time again, since they are rationalised in the exact same way.

  7. Is it possible in dynamic real space axiomatic number 2 can derived particle mass value ? Can numerical axioms as foundation leads to perpetual rate of space as 2.96E+8 ,comprise with elementary particles? Can axiomatic foundation leads to smallest interval 1.344E-51 which perhaps help connect all spectrum in holographic universe? Can Combinatorial math useful in simultaneous state where time is approaching zero instead calculus?

  8. Math is the language of the universe including morality and reality. I believe that there is a profound and fundamental relationship between morals, mathematics and reality and have over the years developed an idea on how that relationship works and how it fits together, but as a layman I do not know what to do with the idea. I stared a blog with a brief overview in a tab at the top of the page. The blog address is johntan34.blogspot.com and the blog name is Revolution of Reality. Any help would be much appreciated and a comment can be left at either of my blogs Revolution of Reality or my Happy Colors blog at johntan33.blogspot.com

    Here are the bullet points to my idea:

    1)There is a moral force in the universe that is governed by natural laws

    2)This moral force is the same through out the known universe and originated with the other known forces, perhaps at the big bang.

    3)Those moral laws are able to be mathematically defined and calculated

    4)There are moral values that are constant and there are other moral values that are variable depending on the circumstance or frame of reference

    5)That frame of reference gives you a perspective and perception which gives you a reality

    6)That reality through that perspective is able to be mathematically defined and calculated

    7)The validity of a moral issue within that defined, or given reality is able to be mathematically defined and calculated following the natural moral laws of the universe