Truth, Provability and the Fabric of the Universe

Here is my talk to the University of Rochester’s Society of Undergraduate Math Students on “Truth, Provability and the Fabric of the Universe”. The audience was great, and except for a couple of slips of the tongue (like “Sir William of Ockham” for “William of Ockham”), I thought it went very well.

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Faithful readers will recognize multiple themes from the book The Big Questions, and from numerous past blog posts, including:

Godel in a Nutshell

Non-Simple Arithmetic

Jellyfish Math

Real Numbers

Just the Facts

First Things and Second Things

Godel, Fermat, Hercules

Basic Arithmetic

Basic Arithmetic, Part II

Basic Arithmetic, Part III: The Map is Not the Territory

Basic Arithmetic: On What There Is

First Things

Berry Interesting

That Does Not Compute

The Number Devil

Simple as ABC

What Would Copernicus Think?

Accounting for Numbers

Many Many Worlds

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18 Responses to “Truth, Provability and the Fabric of the Universe”

1. 1 1 Jack

any way iphone users can easily view this ? I hate flash.

2. 2 2 Philip Maymin

I just watched it on iPhone (it’s very good). Check out eg the Skyfire web browser app.

3. 3 3 Jens Fiederer

Even if he hadn’t been awarded the title at the time, I think the professor was right to so honor him.

4. 4 4 Brian

Steve,

Thanks for posting the talk. I found it interesting and informative. I was also glad to hear you address the MUH again, building on the earlier conversations from your Many Many Worlds post.

With regard to your continued claim that “If statements about a thing are objectively either true or false, then that thing exists,” I see that you’ve added a new caveat regarding the fairy statement. I had said that “Fairies are small humanlike creatures with wings” is an objectively true statement, a claim that you rejected. In the video you still reject it but add a parenthetic comment that “It is perhaps…an objectively true statement that the IDEA of a fairy includes having wings.” So you are making a distinction between statements about the thing itself versus statements about ideas of the thing.

But this distinction won’t work at all, for the simple fact that the only statements we can ever make are about our IDEAS of things, not the things themselves. Whether we are talking about fairies or dogs or electrons or Steve Landsburgs or natural numbers, any statement we make is about our conception of those things, and any deduction we make or property we derive comes from that conception, not from the thing itself. When we make an objectively true statement about something, all it means is that each person’s conception of the thing is consistent with every other person’s conception. But it doesn’t guarantee that the thing exists independent of our conception of it.

If you disagree, I’d be happy to hear why. :)

5. 5 5 Steve Landsburg

Brian:

the only statements we can ever make are about our IDEAS of things, not the things themselves.

But this cannot be true, because, for example, the IDEA of a dog is utterly unlike a dog —- a dog, for example, has a tail, and the ability to bark, and a location in space, whereas the IDEA of a dog has none of these things. So surely it is a very different thing to talk about dogs (“Dogs have four legs”) than to talk about the idea of a dog (“The idea of a dog becomes clear to most children by age three”).

If you disagree, I’d be happy to hear why. :)

I do disagree, and have just said why, but Willard Quine said it first, and better, in his essay “On What There Is”.

6. 6 6 CC

Great talk! The best part was your comparison of Peano’s axioms to group theory axioms.

But you’re just making me beg again: Please do a talk or a series of posts about the relationship between large cardinals and proof systems! The fact that Goodstein sequences grow so fast somehow relates to their unprovability of Goodstein’s Theorem in PA, but I can’t see the connection.

7. 7 7 CC

Hey another thing: You mentioned something about cognitive scientists having some idea of why/how a sufficiently complex system that shuttles around information could become self-aware. I was under the impression that we hadn’t the faintest idea how this was possible. (Reading Daniel Dennett just confirmed that to me!) To what are you referring?

To me, this is more mysterious than why there is something instead of nothing!

8. 8 8 Roger

I downloaded a 645M file, but could not get it to play.

9. 9 9 bigjeff5

Roger, it’s flash video, so you’ll need a player that can play flv’s or a converter.

Wimpy is a cross-platform player, and many other players will as well. You can also get codecs that allow your favorite video player to play them.

For iPhone, it looks like PlayerXtreme HD is the app to get.

For Android it probably worked already, and if not there are dozens of apps that will do the job.

10. 10 10 bigjeff5

Brian,

I think Steve’s point is that the idea of fairies exists, as far as any idea can be a real thing (i.e. it’s stuck in the minds of those who hold that idea), but fairies do not exist. The idea exists because it has objective properties. One of the essential properties of ideas is that they are generated and contained within minds. Another property is that an idea can be copied to another mind without removing the idea from the first mind – though the resulting idea is often slightly different than the original. Another property is that ideas can be stored in a number of forms – text, audio, pictures, videos, computers, etc. – for retrieval by minds or for copying to new minds.

The idea is a separate entity from the thing it represents, and has its own kind of existence. There is no living creature we know of that objectively possesses the properties described by the idea of a fairy (that is, it has those properties regardless of whether or not the idea of a fairy exists), so as far as we know fairies do not exist.

Likewise, the idea of dogs is real, and dogs are real. These are still different, real things, but in this case the idea describes another real thing. The existence of dogs does not rely on the idea of dogs, they are simply described by it. Also, while dogs have the properties described by the idea of dogs, the idea has a lot of properties that dogs do not have. For example, dogs are not generated or contained within minds. They cannot be easily copied. They can’t be stored as text, or audio, or video. Attempting to do these things to a dog will probably kill it (I’m not sure how you’d go about turning a dog into text, but I can’t see it surviving the process), and still not get you a stored dog.

In other words, I can go out and find an animal that matches my idea of a dog and say “Look! It’s a dog! It’s got the properties of a dog!” because dogs exist beyond the idea of dogs. I can’t do that for fairies because they don’t exist.

So the question is whether math exists like dogs, or if it exists like the idea of dogs or the idea of fairies. If math exists like dogs, it has to be fundamental. More fundamental than things like the value of charge in electrons, since it describes those values. If it is more fundamental than the things which make up our universe, then chances are the universe is a layer of existence that sits on top of the more fundamental layer that is math.

It seems to me math exists a whole lot more like dogs than ideas, since even though it shares some properties with ideas it has a lot more properties that ideas don’t have. My confidence on that is not very high, though, as it seems like there is plenty of room to argue.

11. 11 11 Roger

12. 12 12 bigjeff5

Ah, that’s unfortunate. There’s probably a way to repair it, but it tends to take a fair bit of effort, and I’m not even sure flv’s are structured to allow that.

It’s a pretty good talk though.

I’ve found myself looking back through this list of posts and keep getting hung up on the arguments between Steve and Snorri. It’s fascinating, partly because I’m pretty sure I’ve known people who would have exactly the same kind of argument with Steve (and be oh so wrong in the process).

13. 13 13 Steve Reilly

Another mistake early on: Borges didn’t come up with that dialogue between Achilles and the Tortoise. It was Lewis Carroll: http://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles

14. 14 14 Henri Hein

I got a chance to watch this last night and enjoyed it. One question I had:

Is it possible that physicists use math to describe the universe because it is the tool they happen to have available? They also use English to describe the universe, but we don’t think the Universe is made of linguistics (except maybe Chomsky). I know it’s not a good analogy, but cannot think of a better one at the moment.

15. 15 15 Brian

“the only statements we can ever make are about our IDEAS of things, not the things themselves.

But this cannot be true, because, for example, the IDEA of a dog is utterly unlike a dog”

Steve and BigJeff5,

I agree that dogs and the ideas of dogs are very different things, but how does that address, much less refute, the statement I made above?

Here’s why the statements and inferences we make are based on the concept of a thing and not the thing itself. The acts of drawing inferences, deriving properties, and making statements about them occur through well-defined processes in my (or your) brain. Those processes act only on the physical representation of the thing in my brain–the mental concept–and not on the external object. Since any statement you or I make is a straightforward result of those processes, the statements each of us makes reflects our concept of the thing and not the thing itself.

Now in re-reading what I said and what you replied, I realize that my use of the word “about” may be partly responsible for the disagreement. Certainly, we can talk about the dog, as well as talk about our idea of a dog (“I remember when I thought dogs were scary…”), but I didn’t mean to use the word “about” in quite that way. What I meant instead is something like this. When I say that dogs can run, what is really meant is that my mental model (concept) of a dog implies or includes the property of being able to run. This is no different than saying my mental model of a fairy implies or includes the property of being able to fly. It also happens that my conception of dogs includes the state of existence whereas my conceptions of fairies does not.

But the bottom line is that all any of us knows or can comment on is based on our own conceptions of things. To the extent that a given conception is consistent with all our other conceptions (many of which are actively updated through what we call “experience”), we ascribe existence to it.

bigjeff5, you say “the idea has a lot of properties that dogs do not have. For example, dogs are not generated or contained within minds. They cannot be easily copied. They can’t be stored as text, or audio, or video. Attempting to do these things to a dog will probably kill it (I’m not sure how you’d go about turning a dog into text, but I can’t see it surviving the process), and still not get you a stored dog”

and then say “It seems to me math exists a whole lot more like dogs than ideas…” Well, your above description of what we can do with ideas sounds an awful lot like what we do math. So how is it more like a dog?

I do have a number of arguments to make that seem to imply that mathematics merely offers a system isomorphic to the universe rather than being the fundamental nature of the universe itself (ala the MUH), but I’ll have to offer those when I have more time.

16. 16 16 Steve Landsburg

Henri Hein: I take your point, and I cannot dispute it, and I believe it might be exactly right, in which case all of the things I believe are exactly wrong.

One reason, though, that I am skeptical of your viewpoint is that it leaves a big open question: Why is there something instead of nothing?

The mathematical universe hypothesis more or less answers this question: There is something rather than nothing because there is mathematics, and mathematics exists by necessity.

Now a reasonable person could well object that “mathematics exists by necessity” is an assertion that needs further argument. I would, however, disagree with that reasonable person, having given a variety of arguments in my book that seem satisfactory to me.

17. 17 17 Henri Hein

Steve Landsburg: Thanks for the reply. I agree that your explanation does answer the something vs. nothing question. I understand its appeal. There is a neatness to it. I’m still thinking about it, though.

18. 18 18 dave

the open question ‘why?’ – why not?
wouldn’t math better answer the question ‘how?’

we have the power to crack the atom and yet we follow the timeline of a pope from 1000 years ago.

February 29th. does it exist?

any number divided by zero remains undivided and therefore unchanged. 1/0 = 1