Raft to the Future: Unraveling Math Confusion

[malthis]

Hello. It's been a long time since posting. I'm a layman who is intrigued but very confused about something I read recently. This article in Discover Magazine says:

"We now know that mathematical truths cannot fall into predictable patterns and that perfect mathematical systems (ones that are complete and without contradictions) are impossible. We see regularity in the physical universe. Photons arriving from fabulously distant and ancient regions of the universe are fundamentally the same as the photons your eyes perceive as you read this page. Mathematical truths are different. Once you have proved one theorem, you have no indication of what the next one will be like or how hard it will be to find. Mathematical ideas are less regular than reality!"

Does this make any sense to anyone? I get the part about LQG and everything else but I don't know what to make about these math claims. Here's another quote: "mathematics is never complete and always gets weirder the more you understand it" now this seems to go against the common idea that because string theory is so 'elegant' that it must be true in some way. Instead this is saying that math is not reliable the more you probe? I'm not sure what it's saying. Is it chaos theory? Can someone out there clear up this matter? It's driving me crazy.

Here's the whole article
http://www.discover.com/issues/oct-06/departments/jarons-world-raft-future/

Much thamks in advance.

 

[selfAdjoint]

Hello. It's been a long time since posting. I'm a layman who is intrigued but very confused about something I read recently. This article in Discover Magazine says:

"We now know that mathematical truths cannot fall into predictable patterns and that perfect mathematical systems (ones that are complete and without contradictions) are impossible. We see regularity in the physical universe. Photons arriving from fabulously distant and ancient regions of the universe are fundamentally the same as the photons your eyes perceive as you read this page. Mathematical truths are different. Once you have proved one theorem, you have no indication of what the next one will be like or how hard it will be to find. Mathematical ideas are less regular than reality!"

Does this make any sense to anyone? I get the part about LQG and everything else but I don't know what to make about these math claims. Here's another quote: "mathematics is never complete and always gets weirder the more you understand it" now this seems to go against the common idea that because string theory is so 'elegant' that it must be true in some way. Instead this is saying that math is not reliable the more you probe? I'm not sure what it's saying. Is it chaos theory? Can someone out there clear up this matter? It's driving me crazy.

Here's the whole article
http://www.discover.com/issues/oct-06/departments/jarons-world-raft-future/

Much thamks in advance.

The point about "perfect mathematical systems (ones that are complete and without contradictions) are impossible." is an overgeneralization of Goedel's theorem, which abolished Hilbert's program to express all of mathematics as a deduction from a finite set of axioms. Goedel showed that THERE EXIST mathematical theories, a broad and important class of them, those that can derive an axiomatized proof of arithmetic from set theory, that are incomplete approximately as the article states, and Turing showed essentially that they are the same as the things a digital computer can compute.

But there exist other branches of mathematics that don't fall under the shadow of Goedel. Geometry is one of them. Roughly the things an analog computer can compute.

The point that new theorems come as surprises is surely not a defect of mathematics, but a virtue, and the author has much misunderstood physics if he thinks reciting the consequences of an existing theory can be compared to finding a really surprising new theorem. That's as wrong-headed as the idea that research mathematics is about solving existing equations only.

 

[malthis]

Good response. I did a quick study on Goedel and think I have a better grasp on the incompleteness of math. Whether or not you agree with his LQG conclusions, please consider this statement:

"Suppose the universe correlates with some patch of math. That patch cannot be complete and will inevitably bleed into additional math that is even stranger than the starting patch."

Is this an example of the harder you look the less you know because of the nature of a self-referential system? If that's true might it spell doom for the proponents of string theory, who believe their math is so perfect?
P.S. Sorry if this has become so math oriented. I now realize it belongs in the math section and welcome any admin to move it there.

 

[ccdantas]

Malthis puts a very interesting question here.

What do you think of this text by Wigner?

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html"

Thanks,
Christine

 

[selfAdjoint]

Sorry if this has become so math oriented. I now realize it belongs in the math section and welcome any admin to move it there.

Not at all. It is about the application of math to physics, and as you can see from the classic text linked to by Christine above, it is one that has fascinated great physicists as well as those of us who post here. I'll get back to you tomorrow on your questions (and on rereading Wigner's essay after all these years! Thanks Christine!).

But just a hurried top-of-the-head comment: though the "more you know the more understanding recedes" phenomenon is certainly real it applies in all fields not just those involving math. Look at the opinions of deep Hebrew scholars on Genesis versus the man in the street. And the problem there is unlimited complexity, not failure of math because of some internal limitation.

 

[Dcase]

Since the Greek root of physics is "Nature", in the most liberal interpretation, physics includes the science of all nature [even biology].

In a more conservative interpretation, physics might be restricted only to mechanics or electro-mechanics.

[I enjoy your blog!]

 

[paw]

It seems to me that the author, Jaron Lanier, is putting the cart before the horse.

Suppose the universe correlates with some patch of math

All of mathematics is, in one way or another, a tool for conceptuallizing, or modelling, physical reality. At no point does the correspondance of a mathematical model with physical reality necessarily say anything about the underlying mechanism of that reality.

Hooke's Law describes the motion of a spring in terms of applied forces and position but the mechanism is rooted in the electromagnetic interaction.

Rather than the universe correlating to some patch of math, he should have said 'suppose some patch of math describes the universe'. Stated this way any weirdness, supposing there is weirdness, in mathematics is moot. The math describes the system, that's all.

Taking this one step further, our hypothetical, universal model (TOE), may describe the system (universe) adequately, or even perfectly and still have nothing to say about the underlying mechanism. In other words, perhaps our model has 10 physical dimensions. That doesn't mean the universe must also have 10 physical dimensions.

It seems to me more and more theoreticians, amateur and professional, are making this mistake. Assuming that mathematical models are the system rather than just describe the system.

 

[Chronos]

In my admittedly naive perspective, Goedel's incompleteness theorem is to math as Heisenbergs uncertainty principle is to physics.

 

[selfAdjoint]

Suppose the universe correlates with some patch of math. That patch cannot be complete and will inevitably bleed into additional math that is even stranger than the starting patch."

1. As I stated before it is not the case that any randomly chosen "patch of mathematics" cannot be complete. Some can and others can't. So this stiplulation is false.

2. The conclusion that the patch will inevitably "bleed into" additional math that is even stranger assumes that the initial patch was "strange". By whose standards? Irrational numbers are "strange" to some, complex numbers to others. As for bleeding into other mathematics that is a characteristic of creative math; it is connected to all the rest of mathematics at some level.

So I think the passage is describing a real effect, but it is marred by a misunderstanding of the cause of that effect, and a tone of relentless contempt toward anything unusual by his standards. Math is not strange to the mathematicians who do it; consider the enthusiasm of John Baez and Kea for higher category theory, which could fairly be described as strange to an older generation.

 

[Mike2]

In my admittedly naive perspective, Goedel's incompleteness theorem is to math as Heisenbergs uncertainty principle is to physics.

Godel's incompleteness theorem states that you can always find another statement that's true but not provable with that system of math. OK. But what does that have to do with the completeness of physics. We're not trying to find every true mathematical equation. We're trying to find only one math statement from which to predict all of physics. All of physics is not all of math.