To understand the universe at the deepest level, we need to know not only

howthe universe behaves, butwhy.

Why is there something rather than nothing?Why do we exist?Why this particular set of laws and not some other?

So say Stephen Hawking and Leonard Mlodinow in their book The Grand Design, and so say I.

The Big Big Question is the first one: Why is there something rather than nothing? Hawking’s answer: The laws of physics — and especially the form of the law of gravity — allow for the spontaneous creation of universes out of nothing at all. We live in one of those spontaneously created universes. But this, of course, only serves to raise a new Big Big Question, namely: Why are the laws of physics as they are? Hawking’s answer: The laws of physics must be consistent and must predict finite results for the quantities we can measure. It turns out that those criteria pretty much dictate the form of the laws of physics.

So unless I’ve misunderstood him, here is Hawking’s position: In order for us to be able to measure the things that we measure, the laws of physics must have a certain form, and in order for them to have that form, universes must be able to arise from nothing. Therefore our universe was able to arise from nothing. But this does not seem to answer the question of why things couldn’t have been very different. Why couldn’t there have been no us, no measurements, no laws of physics and no anything?

I know of only one satsfying (to me) answer to this question, and Hawking comes tantalizingly close to it without ever quite going there. He spends a lot of pages reviewing current physical theories but never mentions the one glaring feature they all share: **Every modern physical theory, taken literally, predicts that our universe is a mathematical object**. For example, the simplest version of special relativity posits that we live in a four-dimensional geometric object called “spacetime”. More sophisticated theories posit that spacetime is part of some larger geometric object whose properties we perceive as “forces” or “particles”. According to modern physics,

**everything is made of math**.

Now you might say that physical theories aren’t meant to be taken **that** literally; that instead they describe mathematical objects with properties that are **analogous** to the properties of the physical universe. But it seems to me that if, like Hawking, you trust in theories to explain the mystery of creation itself, then you ought, at least provisionally, to take those theories literally. Otherwise, what you’ve got is not a theory. It’s a theory plus a bunch of *ad hoc* and arbitrary choices about which parts of that theory you choose to believe.

Once you believe the universe is a mathematical object, its existence ceases to be a mystery—at least if you believe, along with most mathematicians, that mathematical objects can’t help but exist. Hawking embraces M-theory, which tells us that the universe is a particular 11-dimensional object (with a whole bunch of additional geometric curlicues that appear to our senses as everything from stars to bacteria. M-theory also says there are a whole bunch of other 11-dimensional universes, all of which were spontaneously created, and we just happen to live in this one.

What I’m suggesting is that the universes of M-theory are only a tiny fraction of the universes out there, because anything that exists mathematically is a universe, though most of them (like most of the universes of M-theory) are far too simple to contain anything like sentience. This is essentially the view of cosmologists like Max Tegmark of MIT.

Hawking is 90% of the way there. The many universes of M-theory are mathematical objects, and all are pieces of a bigger mathematical object called the multiverse. “Spontaneous creation” means that the multiverse is structured in such a way that it must contain these universes. But why is there a multiverse and why is it structured in that way? That’s the part Hawking seems not to address. Proposed answer: The multiverse itself is only one of many multiverses. They all exist for the same reason the natural numbers exist: The laws of mathematics require it. And unlike the laws of physics, which differ from multiverse to multiverse, the laws of mathematics, which live outside any universe, could not have been otherwise.

(For more on this subject, read Chapter 1 of *The Big Questions* !)

Something I mentioned in email when I read this argument in The Big Questions:

This theory could be true, but consider the moral implications. Every state of every universe that could exist, does exist (since it’s a mathematical structure). Therefore, there’s no moral imperative to save someone else’s life, because no matter what you do, a universe continues to exist in which their life was saved, and a universe continues to exist in which it wasn’t. The total amount of suffering of other beings is the same in either case.

You could only justify saving your friend’s life on the selfish grounds that *your* conscious mind then gets to continue living in a universe where they were saved. And even that justification doesn’t work for taking an action that saves the life of a poor child in Africa or someone else that you’ve never met.

Ironic, given the amount of time you’ve spent advocating that we should give equal weight to the concerns of faraway strangers as to our fellow countrymen — that your cosmological argument would undercut that :)

Are you saying that everything possible exists?

Care Bears and unicorns and Jesus are also mathematical objects. Do they exist, too?

Bennett Haselton: Even if every outcome that could happen does happen in some universe of a multiverse, it does not follow that all the outcomes will be weighted equally. Imagine that a pair of 6-sided dice are cast and, for simplicity, 36 universes result. There would be more universes where the dice came up showing a total of 7 than a total of 2. And that’s the way to bet.

If the splitting of universes in your example is quantized, then a probabilistic likelihood that you will make good moral choices might well result in reduced suffering summed over all the universes in which you exist. If there are countably many universes, or even a continuous wave function of universes as in good old many-worlds quantum theory, I think it’s still reasonable to suppose that more probable events are more prevalent.

So keep on making good moral choices. It’s worth it.

In my opinion, the problem with this idea is that in the past physical theories have always proved to be approximations to a more accurate theory. There is no reason to think that the current crop of theories — even M-theory — is the final real McCoy. I’d say it makes no sense to assert that the real world is a mathematical object that only approximates it’s known behaviour.

Is it possible that the universe is a mathematical object as yet undiscovered? I can’t refute that. But I understand that Godel’s theorems demonstrate that a mathematical formalism is necessarily incomplete. Is the universe a mathematical object taken from such a formalism? This doesn’t appeal to me very much. I’d rather imagine that our theories will always be approximate, and that the really real world will always remain just out of reach.

Why couldn’t the laws of mathematics have been otherwise?

Do different axiom choices result in different mathematics or are they all still “the same” from your perspective? What about different methods of deduction?

Perhaps the laws of physics and mathematics are as they are because that’s the best way human beings can currently understand them.

There are good reasons why Hawking didn’t say this, and I had to chuckle a bit when you suggested that the man who possibly knows more about the universe than any other was “about 90% of the way” to understanding the universe the way an economist does. :p

What you’re saying (in my view) is basically that Italy is shaped like a book because that’s the way we drew the map. It must be shaped like a boot because otherwise our map would be inconsistent. Therefore, Italy is a map.

Human beings developed physics and math to explain the universe in which we exist. That doesn’t mean the universe is an object of the tools we use to explain it.

And now you have done what I knew you would – conflated “exists” in the mathematical sense with “exists” in the concrete sense, without really saying what you mean by either one.

Yes, its a lovely looking structure, philosophically. That doesn’t make it true. Whatever “true” means.

And don’t forget that infinities are different sizes.

I wonder what Paul Krugman’s thoughts are on this.

Could someone do me a favor and provide a brief explanation of how M-theory says something can come from nothing?

Aren’t there a lot of mathematical objects that don’t exist in the physical world? For example, a perfect circle. I don’t see why M-theory requires that our universe exist any more than the notion of a circle requires actual objects shaped like circles to exist physically.

Ask a question that can be answered only with a tautology and you get a tautology.

Jonathan Campbell: The perfect circle is a mathematical object and therefore counts as a (very simple) universe in its own right. Our universe is a much more complicated mathematical object. They exist separately. We don’t expect one to exist *inside* the other.

Jonathan Campbell: In grade school, I was taught that medieval astronomers thought perfect circles must exist, and invented the notion of epicycles to support their faith in the perfect circle. Now used as a metaphor for bad science, epicycles are actually a pretty good computational gimmick.

So M-theory is composed of a series of axioms, and one of them must be along the lines of “This theory does not exist merely as an abstraction. It has a physical manifestation.” Let’s say you strip that axiom away from M-theory, and call the pared-down theory “M-theory (Beta).” Surely the Beta version exists just as well as the full version. It seems to me that in adding the abovementioned axiom to the Beta version, to get to the full version, you are just gratuitously allowing math to encroach on physics.

This reminds me of Anselm’s ontological argument for the existence of God (http://en.wikipedia.org/wiki/Ontological_argument):

1. If I am thinking of the Greatest Being Thinkable, then I can think of no being greater

1a. If it is false that I can think of no being greater, it is false I am thinking of the Greatest Being Thinkable

2. Being is greater than not being

3. If the being I am thinking of does not exist, then it is false that I can think of no being greater.

4. If the being I am thinking of does not exist, then it is false that I am thinking of the Greatest Being Thinkable

It seems to me that both Anselm and you/Hawking think you have discovered objects which exist in your imagination (Anselm: Greatest Being Thinkable; you: M-theory) which, as a result of their properties, require the existence of some physical things (Anselm: God; you: multiverses).

(Of course Anselm has a steeper hill to climb since he is trying to prove the existence of the physical object, whereas you are only trying to explain it.)

Steve, it sounds like you believe in Simulism or the Simulation hypothesis. There is a long history of such beliefs, but I doubt that Hawking would endorse them.

Jonathan Campbell – To Roger Schlafly’s point, I don’t recall Hawking ever explicitly endorsing such things.

Not to cite my own example, but you can basically break it down like this:

1. Italy is shaped like a boot because that’s the way we drew the map [false]

2. Italy must be shaped like a boot because otherwise our maps would be inconsistent [entirely true!]

3. Italy is a map. [false]

Hawking makes Statement #2 in his book, which is objectively true. M-theory is consistent with everything we can measure about the universe, so it’s basically as much as we know about the universe right now.

However (in my opinion) Landsburg is making Statements #1 and #3, both of which I disagree with.

“Hawking makes Statement #2 in his book, which is objectively true. M-theory is consistent with everything we can measure about the universe, so it’s basically as much as we know about the universe right now.”

Nice, except that the M-theory map also says there are italys that are shaped like gloves, trousers, fedoras…

Although I don’t have a physics background, everything I’ve read seems to suggest that you, Tegmark, and all the Many-worlders are basically right. I’m still not sure what it would mean for all mathematical objects to truly exist though. Do they require a substrate to exist on? If you could describe a mathematical object with sufficient accuracy, is that all you’d need to instantiate it? If not why not? And does this mean that all possible patterns exist somewhere/somehow? What would this mean regarding travel between parts of the multiverse or to different universes? It’s a lot to wrap your head around.

I’m surprised someone would find this a satisfying explanation. “Everything is made of math” doesn’t seem to have any real meaning.

Quoting from Good and Real by Gary Drescher,

“At this point, it behooves us to step back from the question Why is there something rather than nothing, and to rearrange the punctuation a little: why, is there something rather than nothing? Is there really? Or, more to the point: what, if anything, would be the difference between there being something and there being nothing? What difference, if any, does the universe’s existence make?

The question sounds odd. Of course it makes a difference—it makes literally all the difference in the world. We know the universe exists because we see and feel its myriad constituents. If the universe didn’t exist, there would just be nothingness instead, and we wouldn’t be wondering about it. It seems, then, that whatever equations describe our universe are somehow endowed with a ‘spark of existence’ that gives substance to the otherwise vacuous equations.

…

… imagine an alternative set of equations that define an alternative, imaginary universe in which evolve intelligent, inquisitive beings like us. If we could compute what unfolds from those equations and watch what the eventually evolved beings say when they contemplate their world, presumably we would not then find them lamenting that their universe, for all its grandeur, unfortunately lacks that all-important spark of existence! On the contrary, of course, their universe looks and feels to them as obtrusively, overwhelmingly real as ours does to us—and we would see them think so and say so.

Most importantly, they would think and say so for the same sort of reason as we do, a reason that must be rooted in the equations themselves (because the equations themselves ultimately specify every detail of those thoughts and words), without recourse to any spark of existence. And even if we did not carry out the computation of what the alternative equations specify—even if those equations were left out in the cold, unnoticed and unexamined—those equations would still be specifying a universe in which intelligent beings perceived and spoke of what they thought is a spark of existence, just as we do, and for the same reasons.”

What are the mathematical foundations of morality?

Trevor: Thanks for this provocative quote.

I believe that Mr. Drescher and I are saying the same thing in different words. He imagines a universe that is mathematically well specified but fails to exist. I choose to use the word “existence” in a way that encompasses anything that’s mathematically well specified. So we’re talking about exactly the same thing, though he chooses to call it “non-existent” and I choose to call it “existent”. In either case, the point is that its nature is purely mathematical, and that a purely mathematical universe can contain sentient intelligent beings such as ourselves.

Ryan:

1. Italy is shaped like a boot because that’s the way we drew the map [false]…

(in my opinion) Landsburg is making Statements #1 and #3I don’t see where you’re getting this.

Jonathan Campbell:

So M-theory is composed of a series of axioms, and one of them must be along the lines of “This theory does not exist merely as an abstraction. It has a physical manifestation.” Let’s say you strip that axiom away from M-theory, and call the pared-down theory “M-theory (Beta).” Surely the Beta version exists just as well as the full version. It seems to me that in adding the abovementioned axiom to the Beta version, to get to the full version, you are just gratuitously allowing math to encroach on physics.I don’t see what this could possibly mean. M-theory, like every other modern physical theory, describes a purely mathematical structure and says “that’s the universe”. If you delete the phrase “that’s the universe”, then your theory is pure math with no physical content. If you preserve the phrase “that’s the universe”, then your theory predicts that the universe is a purely mathematical object. There’s no in-between.

damm. i hate when i misclick my mouse and lose an entire train of thought. even if that train is on a closed track.

@ryan dr. landsburg has a phd in math, no? i think he moonlights in economics because nobody listens to mathematicians except for other mathematicians and physicists.

it doesnt surpise me at all that a physicist would imagine the universe as a mathematical object.

why art? why beauty? why music? why poetry?

i spotted this poem on myspace doc. i clicked on the ‘most popular blog’ link.

http://blogs.myspace.com/index.cfm?fuseaction=blog.view&friendId=77178284&blogId=539392069

even if we ever figure out how many dimensions our universe consists of, the human condition will still be one of abject isolation. our giant monkey brains can split atoms. we may someday be able to smoosh them together. we will still be monkeys..all alone..no one to cheer for us except ourselves.

i really like the title of his book. if i were someone that believed in intelligent design i would most certainly buy a copy.

Roger Schlaffy: yes, it is hard to avoid the conclusion that the larger infinity of universes must be simulations. For each “real” universe can contain an infinite number of simulations.

Prof. Landsburg –

What I’m getting at is that, contrary to the claims in your book, my position is that human beings invented mathematics. I consider mathematics the “map” we use to understand the universe (Italy). It’s a systematic human thought process designed to make difficult concepts a priori better understood. It’s an amazing human accomplishment, but it is a human accomplishment.

So, my analogy is that your claim that mathematics predates human beings and exists outside of them is akin to proclaiming that Italy is shaped like a boot because we drew the map that way. Your claim that the universe is basically made of mathematics is akin to proclaiming that Italy *is* a map.

Italy exists (we assume) independant of the map. Humans use the map to describe Italy. However, we already know that Italy exists, so we know which is the correct interpretation.

Perhaps a better example than Italy would be a diagram of machine. In interpretation (A) the diagram is a plan of an existing machine. In interpretaion (B) the diagram is the blueprint for the machine. In (A), the diagram is only the way it is because of the machine.

In (B), the machine is only the way it is because of the diagram.

If we had in front of us the diagram and the machine, how could we tell which was the correct interpretation?

Steve: I agree with everything you’ve just said in that comment. And I would just add that adding the phrase “that’s the universe” does nothing to explain *why* that is the universe, which was our goal. I know you would argue that “that’s the universe” is a mathematical fact, but to me that step requires using a different definition of “math” than the one we are used to.

We are puzzling over whether the universe (i.e. everything) is made of math or not. I am curious – what, other than the *why* of the universe, would be substantively different between a universe that is said to be made of math and one that is not? I don’t believe there would be any substantive difference.

@Ryan, there’s a fundamental difference between a map being a representation of Italy, and math being a representation of physics. When we draw a map of Italy, we have foreknowledge of the shape of Italy, and draw the map to correspond as closely as possible to that shape.

Math was created in parallel to physics, and turns out to be an amazing model of our physical universe, even to the extent that almost every concept in math has an analog in physics. However, math wasn’t designed to represent physics.

You could use the map of Italy as a metaphor for math if someone had drawn a map of an imaginary place, and then discovered that the map actually represented Italy.

My perspective on the relationship between math and physics correspond to Steve’s. Math represents our physical universe just too neatly for it to be a coincidence. For example, take acceleration. x = at^2, because a = dv/dt. But why? If the universe weren’t made of math, then these equations wouldn’t have to be so neat. We might have x = at^1.35319, for example.

Jonathan Campbell: Max Tegmark has emphasized that the “made of math” hypothesis does, in principle, have testable predictions. I might blog about this soon.

Ryan: Except that my claims are not what you say they are because I do not proceed from the assumption that mathematics is a human invention.

Prof. Landsburg – Understood. I think there is value in appreciating the finer points of both perspectives, which is why I enjoyed this blog post and the corresponding sections in your book, even though my own conclusions are a little different.

It is a fascinating thought experiment to consider the ramifications of either view. In particular, I think your view of mathematics in this regard greatly shapes other aspects of the ideas in your book. For example (even though I might not be able to explain exactly why – although you probably could!), I see your view of mathematics being a major driver of your preference of consequentialist ethics.

Between your blog and your book, you’ve really gotten my gears cranking on this, which is much appreciated. :)

Ryan: Thanks for being part of this, and for sharing your insights. I’m glad you’re here.

I look forward to the blog post about the testable predictions.

Steve: it seems to me that your theory suggests that there could theoretically be a person so smart that he could be omniscient, i.e. that he could know everything about the universe, and that he could predict the future. If all facts of the universe are mathematical facts, then shouldn’t an incredibly smart person be able to derive them? Or is there a category of mathematical facts that cannot be derived a priori?

And how does this jive with quantum mechanics, which suggests that there is true uncertainty? Would an incredibly smart person be able to derive only the starting wave functions that described the origin of the universe (which might not be subject to quantum uncertainty), or would he also be able to predict the outcomes of subsequent measurements (which are subject to uncertainty)? If only the former, then in fact there is some category of mathematical facts (results of measurement) that are not derivable even in theory. If the latter, then the standard interpretation of quantum mechanics is not valid.

Jonathan:

We know for certain that no algorithm (and hence presumably no sentient being) can derive all of the facts about any sufficiently complex mathematical object. For example, no algorithm can derive all of the facts about the natural numbers.

Re your second paragraph, it is only certain interpretations of quantum mechanics, not quantum mechanics itself, that incorporate true uncertainty.

Fine, so Godel’s imcompleteness theorem says that we shouldn’t expect all facts to be derivable, i.e. establishes a maximum on what we can know about a mathematical object. Is there a theorem that establishes a (non-null) minimum? Or is it possible for there to be mathematical objects about which nothing can be discovered? If there is a non-null minimum, what should we expect that a sentient being or algorithm would be able to derive about the universe?

Ond 2nd point: does your hypothesis force us to accept an interpretation of quantum mechanics that rejects uncertainty?

Jonathan Campbell: Godel’s theorem does not put any limit on the facts about a structure that can be derived. It puts a limit on the facts that can be derived by *any given algorithm*. Any fact can be derived by some algorithm or other. But no single algorithm can discover all facts.

Any thoughts on cellular automaton?