A "Harmonious" Path
June 17, 2008 3:35 PM Subscribe
Accidental Astrophysicists: "They started with algebra and ended up learning about gravitational lensing (PDF)." [Via linkfilter]
“I’ve been working on math all my life, and this was a once-in-a-lifetime experience, where I found out that something I did was relevant in a totally unrelated field.”
FTFY
I'm sure if pure mathematicians and physicists ever really sat down and compared all their notes, they'd find a LOT of reinvented wheels and problems that have already been solved. Normally they only find these overlaps when the problem obtains epic status (such as Fermat) and people outside the subsubsubfield hear about it.
posted by DU at 4:19 PM on June 17, 2008 [2 favorites]
FTFY
I'm sure if pure mathematicians and physicists ever really sat down and compared all their notes, they'd find a LOT of reinvented wheels and problems that have already been solved. Normally they only find these overlaps when the problem obtains epic status (such as Fermat) and people outside the subsubsubfield hear about it.
posted by DU at 4:19 PM on June 17, 2008 [2 favorites]
Avenger: the debate rages in the form of discovery versus invention.
My own intuitive, uninformed take on it is that we discover the logical consequences of the axioms we set forth in the creation of a system. For instance, take i; it's an invented solution to the problem of the value of the square root of -1, but Euler discovered later that e to i * pi = -1. The rules of how these symbols operate create these relations that we later find to be true, it seems.
Where that framework of consequences appears from, and how it relates to the universe at large, is still a mystery, but I can't convince myself that it's all an invention.
posted by invitapriore at 4:45 PM on June 17, 2008 [2 favorites]
My own intuitive, uninformed take on it is that we discover the logical consequences of the axioms we set forth in the creation of a system. For instance, take i; it's an invented solution to the problem of the value of the square root of -1, but Euler discovered later that e to i * pi = -1. The rules of how these symbols operate create these relations that we later find to be true, it seems.
Where that framework of consequences appears from, and how it relates to the universe at large, is still a mystery, but I can't convince myself that it's all an invention.
posted by invitapriore at 4:45 PM on June 17, 2008 [2 favorites]
Avenger writes "So, does Math have an intrinsic relationship with Reality, or do our brains (reality) just happen to accurately interpret the rest of reality using Math?"
Math=reality. This universe and it's sibling universes are substrings of pi.
posted by mullingitover at 5:18 PM on June 17, 2008 [2 favorites]
Math=reality. This universe and it's sibling universes are substrings of pi.
posted by mullingitover at 5:18 PM on June 17, 2008 [2 favorites]
So, does Math have an intrinsic relationship with Reality
Could we come up with types of math that have no relationship with reality? Yes, and I'm pretty sure we have, non euclidean geometry for example. But, many types of mathematics were created because they were thought to relate to the physical world.
Take calculus for example, it's a perfectly self-consistent system that you can use to solve imaginary problems, but it was created to try to solve real problems. But it dosn't map perfectly with reality, because reality isn't composed of infinitely dividable material.
So you can use calculus to figure out the volume of a vase, but you can't get it exactly right.
On the other hand, mathematics exists in our minds, which exit in reality. So all math exists inside reality.
People often say that mathematics is 'universally true' (maybe that's not the exact term they'd use) but ironically they mean that it's true outside of the universe, or in any imaginable universes. But if you are outside of the universe, then there is no one around to appreciate it.
So to sum it up, math exists outside the universe, but all imaginary universes exist inside people's imaginations, which exist inside the universe.
Math was created by people to describe the world, but people were created by the world, through (the mathematically well defined) process of evolution.
posted by delmoi at 6:07 PM on June 17, 2008
Could we come up with types of math that have no relationship with reality? Yes, and I'm pretty sure we have, non euclidean geometry for example. But, many types of mathematics were created because they were thought to relate to the physical world.
Take calculus for example, it's a perfectly self-consistent system that you can use to solve imaginary problems, but it was created to try to solve real problems. But it dosn't map perfectly with reality, because reality isn't composed of infinitely dividable material.
So you can use calculus to figure out the volume of a vase, but you can't get it exactly right.
On the other hand, mathematics exists in our minds, which exit in reality. So all math exists inside reality.
People often say that mathematics is 'universally true' (maybe that's not the exact term they'd use) but ironically they mean that it's true outside of the universe, or in any imaginable universes. But if you are outside of the universe, then there is no one around to appreciate it.
So to sum it up, math exists outside the universe, but all imaginary universes exist inside people's imaginations, which exist inside the universe.
Math was created by people to describe the world, but people were created by the world, through (the mathematically well defined) process of evolution.
posted by delmoi at 6:07 PM on June 17, 2008
DU: Scientists often re-invent things which were long known. Here we review these activities as related to the mechanism of producing power law distributions, originally proposed in 1922 by Yule to explain experimental data on the sizes of biological genera, collected by Willis. We estimate that scientists are busy re-discovering America about 2/3 of time. - Simkin & Rowchowdhury
posted by edd at 6:21 PM on June 17, 2008
posted by edd at 6:21 PM on June 17, 2008
Mathematicians are cooks who come up with great recipes. But it's up to the scientists to figure out what the ingredients are.
posted by Citizen Premier at 6:24 PM on June 17, 2008 [2 favorites]
posted by Citizen Premier at 6:24 PM on June 17, 2008 [2 favorites]
I would say just that Math strives to be consistent because Mathematicians fancy consistency for its own sake, and, happily, we've discovered in recent years that reality turns out to be consistent, because it just is that way. Since the set of consistent behaviors is actually pretty small, relative to all possible behaviors, both are drawing from the same pool, and there ends up being a lot of overlap.
posted by blenderfish at 6:45 PM on June 17, 2008 [2 favorites]
posted by blenderfish at 6:45 PM on June 17, 2008 [2 favorites]
Speaking of ingedients, what are the plasma generating capacities of massive bodies?
posted by RoseyD at 6:47 PM on June 17, 2008
posted by RoseyD at 6:47 PM on June 17, 2008
The relationship between mathematics and the natural sciences is neither a new nor solved problem. I remember being amazed that the universe seemed to know about Hilbert spaces, and that waves on water knew that they were made up of a linear combination of sine curves.
It still amazes me actually, the crazy schemes that mathematicians dream up (discover?) that happen to be some deep feature of the universe...
posted by claudius at 6:48 PM on June 17, 2008
It still amazes me actually, the crazy schemes that mathematicians dream up (discover?) that happen to be some deep feature of the universe...
posted by claudius at 6:48 PM on June 17, 2008
@ Avenger
"So, does Math have an intrinsic relationship with Reality, or do our brains (reality) just happen to accurately interpret the rest of reality using Math?"
http://arxiv.org/pdf/0704.0646v2
posted by yoyo_nyc at 6:53 PM on June 17, 2008
"So, does Math have an intrinsic relationship with Reality, or do our brains (reality) just happen to accurately interpret the rest of reality using Math?"
http://arxiv.org/pdf/0704.0646v2
posted by yoyo_nyc at 6:53 PM on June 17, 2008
For instance, take i
For you electrical engineers out there, he means "for instance, take j."
posted by ZenMasterThis at 7:15 PM on June 17, 2008 [6 favorites]
For you electrical engineers out there, he means "for instance, take j."
posted by ZenMasterThis at 7:15 PM on June 17, 2008 [6 favorites]
From Claudius' wiki:
A different response, advocated by Physicist Max Tegmark (2007), is that physics is so successfully described by mathematics because the physical world is completely mathematical, isomorphic to a mathematical structure, and that we are simply uncovering this bit by bit.
In this interpretation, the various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of certain aspects of more complex mathematical structures.
In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics.
Hmm...
posted by Avenger at 7:23 PM on June 17, 2008
A different response, advocated by Physicist Max Tegmark (2007), is that physics is so successfully described by mathematics because the physical world is completely mathematical, isomorphic to a mathematical structure, and that we are simply uncovering this bit by bit.
In this interpretation, the various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of certain aspects of more complex mathematical structures.
In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics.
Hmm...
posted by Avenger at 7:23 PM on June 17, 2008
i is j and j is i and we are all together
posted by lukemeister at 7:27 PM on June 17, 2008 [3 favorites]
posted by lukemeister at 7:27 PM on June 17, 2008 [3 favorites]
Within mathematical theories of reality is the informational ontology. This is pleasing because while it would require an infinite number of bits to naively encode all the digits of pi, and this is therefore impossible, pi does not, in a certain sense, contain an infinite amount of information as one may describe various methods to compute it (a form of compression) in a finite amount of bits. While all those require outside information to understand and use, they are clearly understandable and usable by humans or computers with a finite amount of information.
Recently I've been wondering how the universe seems to compute pi to infinite decimal places instantaneously, or does it?
posted by TheOnlyCoolTim at 7:54 PM on June 17, 2008
Recently I've been wondering how the universe seems to compute pi to infinite decimal places instantaneously, or does it?
posted by TheOnlyCoolTim at 7:54 PM on June 17, 2008
Could we come up with types of math that have no relationship with reality? Yes, and I'm pretty sure we have, non euclidean geometry for example
Actually, the geometry of spacetime is non-euclidean. It's euclidean geometry has no relationsip with reality. :)
My take on this: Mathematics involves postulating objects with certain properties, and then goes on to rigorously prove relationships between those objects, provided they exist. Some of these objects are "invented" to describe the real world: ordinary numbers describe amount of stuff, distances, etc. very well. Planes, lines, points, etc are similar to the basic features we see in the world around us. The science leap occurs when we assume that some real-world object is similar to (or the same as) some mathematical object, and then apply the results of mathematics to predict the relationships between real-world objects. If we knew that there was a 1-1 correspondence, there would be no leap whatsoever, and no mystery to speak of. In fact, there is never any point where we know for certain that the correspondence is correct (are position coordinates scalars, or could they be matrices, or something else?). The success at using math to describe reality shows that we've been reasonably successful at extracting/guessing the most salient properties of the objects in the world around us, and developing the math to relate them. At least, we've been successful when we compare it to the alternative approach, which is "puling things out of your ass".
posted by Humanzee at 8:04 PM on June 17, 2008 [1 favorite]
Actually, the geometry of spacetime is non-euclidean. It's euclidean geometry has no relationsip with reality. :)
My take on this: Mathematics involves postulating objects with certain properties, and then goes on to rigorously prove relationships between those objects, provided they exist. Some of these objects are "invented" to describe the real world: ordinary numbers describe amount of stuff, distances, etc. very well. Planes, lines, points, etc are similar to the basic features we see in the world around us. The science leap occurs when we assume that some real-world object is similar to (or the same as) some mathematical object, and then apply the results of mathematics to predict the relationships between real-world objects. If we knew that there was a 1-1 correspondence, there would be no leap whatsoever, and no mystery to speak of. In fact, there is never any point where we know for certain that the correspondence is correct (are position coordinates scalars, or could they be matrices, or something else?). The success at using math to describe reality shows that we've been reasonably successful at extracting/guessing the most salient properties of the objects in the world around us, and developing the math to relate them. At least, we've been successful when we compare it to the alternative approach, which is "puling things out of your ass".
posted by Humanzee at 8:04 PM on June 17, 2008 [1 favorite]
For instance, take i
For you electrical engineers out there, he means "for instance, take j."
It's long been my firm belief that, in fact, i = -j.
posted by flabdablet at 3:14 AM on June 18, 2008
For you electrical engineers out there, he means "for instance, take j."
It's long been my firm belief that, in fact, i = -j.
posted by flabdablet at 3:14 AM on June 18, 2008
it is pretty clear that nature has mathematical properties. But what is really hard to make sense of is how maths is real without postulating the mind-independent existence of mathematical entities floating about in platonic heaven. I favour an approach where these properties are somehow built into what it is to be a physical substance at all, though call me in 10 years when I've figured out how that works.
But anyway, if the universe is mathematical, it at least would help to explain why things are the way they are. It becomes logically necessary. We then have to get over the idea that M. Night Shyamalan's latest film is logically necessary. And also I really don't see yet how you can get things like temporal direction in a mathematical universe.
posted by leibniz at 5:02 AM on June 18, 2008
But anyway, if the universe is mathematical, it at least would help to explain why things are the way they are. It becomes logically necessary. We then have to get over the idea that M. Night Shyamalan's latest film is logically necessary. And also I really don't see yet how you can get things like temporal direction in a mathematical universe.
posted by leibniz at 5:02 AM on June 18, 2008
...I favour an approach where these properties are somehow built into what it is to be a physical substance at all...
posted by leibniz
Heh.
posted by synaesthetichaze at 7:20 AM on June 18, 2008
posted by leibniz
Heh.
posted by synaesthetichaze at 7:20 AM on June 18, 2008
I've quoted GEP Box once, and I'll quote him again: "All models are wrong; some are useful." Mathematics is a system of logic. Mathematics can be used to model the world. Each such application is an approximation. Always. But some of these approximations are useful.
posted by Mental Wimp at 11:30 AM on June 18, 2008
posted by Mental Wimp at 11:30 AM on June 18, 2008
This has been what I've believed all along, to a great degree. It's not quite Newtonian clockwork, or necessarily Wolfram's cellular automata, but in so much as physics is the hopefully accurate predictive, quantitative modeling of the Universe, it looks as if the whole thing sits on a copy of MatLab, having mathematical rules applied to some currently-defined quantities, over and over again.
To me, the only question has been - precisely what constants are arbitrary or otherwise do not emerge out of the math? Did the universe have any choice in the fine structure constant, or does that number simply evolve out of the math in a non-obvious manner, the way Feigenbaum's constant does? A similar question emerges with whatever physical laws we end up with, unification or not - did those emerge from the math, too, or could we have had some kind of other force simply appear, like, I don't know, another form of electric charge, but this one only repulsive and always at short distances?
posted by adipocere at 2:42 PM on June 18, 2008
To me, the only question has been - precisely what constants are arbitrary or otherwise do not emerge out of the math? Did the universe have any choice in the fine structure constant, or does that number simply evolve out of the math in a non-obvious manner, the way Feigenbaum's constant does? A similar question emerges with whatever physical laws we end up with, unification or not - did those emerge from the math, too, or could we have had some kind of other force simply appear, like, I don't know, another form of electric charge, but this one only repulsive and always at short distances?
posted by adipocere at 2:42 PM on June 18, 2008
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posted by Avenger at 4:15 PM on June 17, 2008