This post over at Thought Mesh raises to related philosophical questions that are fun to think about. The first is the nature of mathematics, which I'll discuss in this post, and the second is the nature of science (or the problem with Einstein), which I'll discuss in a post to come.
Alternate universes, the subject of AOG's post, is an example of a set of scientific speculations (time travel is another) that is allowed, but not compelled, by our current models of reality. One question this raises is the reliability of our models, which, in turn, raises the question of the nature of mathematics.
Is mathematics inherently tied to the nature of the universe, or is it just a really good modeling tool that, if done right, has not yet been inaccurate in our experience. Put another way, does putting two objects down next to two other objects have to result in four objects, or does it just so happen to work out that way?
We tend to assume that mathematics is inherently tied to the nature of reality for two reasons. First, as I've already said, because our experience doesn't include an example of where it doesn't. Second, because math has been usefully predictive. (Are these actually two different reasons, or just separate aspects of one reason? It is in any event useful to discuss them separately.)
The first reason seems compelling -- or, rather, feels compelling -- but is obviously problematic. We have, in fact, a rather small sample of different situation in which math has been a useful modeling tool -- those that can be experienced on Earth, with the tools available to us, over the last 10,000 years. How many different types of situations have we really been in, compared to the age and size of the universe. Plus, we're by nature connection seeking, satisficing animals. Once we've experienced an event, the fact that we can go back and show that the math works isn't really all that interesting. We can always find connections after the fact (connection seeking) and once we've found one tend to stop looking (satisficing). So that fact that mathematics conforms to our experience, as we've experienced it, might not be great evidence about the nature of mathematics.
The second reason, because it avoids the problem of looking backwards, seems the stronger. If we can use math to model unknown facts about the universe, and then test to conform them (Popper looks around triumphantly) then it is that much more likely that mathematics is inherently tied to the nature of the universe. But this, too, proves less than it seems. Because we also do make predictions based on mathematics that, when we test them, are disproved -- otherwise, we wouldn't bother to check, would we. In other words, science, in the Popperian sense, requires that we be skeptical that mathematics, or at least our ability to do mathematics, does accurately mirror reality. We don't accept our models until we confirm them.
So it seems that, speaking of Popper, our intuitive sense that mathematics is inherently tied to the nature of reality is unfalsifiable.
Does this matter? Actually, it seems to matter quite a bit. The rocket-sciences who gravitated to Wall Street over the last 15-20 years made billions with models of the financial market used to value options and derivatives without a history of performance or any tangible, real world, analog. The mathematicians and their banks all assumed that their mathematical models were tied to some underlying reality; that the constructs "really" had the value that the models suggested. We know how that turned out.
Now we're about to engage in an even more drastic economic experiment based on the assumption that global climate models can reliably reflect the future, a proposition that, to the extent it can be falsified, has been. Moreover, we not only have to consider whether mathematical models can model future climate, but also whether the people performing the modeling are doing so honestly and competently. Of course, every time you fly, you're also betting your life (albeit with much better evidence) that there is some relationship between mathematics and reality.
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12 comments:
Interesting post. I'm glad to see you are getting your priorities straight again.
The mathematicians and their banks all assumed that their mathematical models were tied to some underlying reality; that the constructs "really" had the value that the models suggested.I think it is worth noting that mathematical models are abstractions of reality, not reality itself.
The reason we check them is not to see if the math is correct, but rather whether the abstraction is.
Once upon a not too distant past, aeronautical engineers proved that bumblebees could not fly.
Yet fly they did.
The problem was not with the math, but rather the abstraction of reality.
The engineers relied upon laminar airflow.
Bumblebees do not.
It's not so much getting my priorities straight as the semester being over. With luck, my course work is done.
What I'm asking in the post is whether, if we're absolutely sure that something is true mathematically, are we also sure that it is true "in the world." My concern is that we can always find, after the fact, some excuse like "laminar air flow" to convince ourselves that the problem was us, rather than with math.
Well, from my selfish point of view, when you study at the expense of posting, your priorities are inverted.
++++
From your post:
We have, in fact, a rather small sample of different situations in which math has been a useful modeling tool -- those that can be experienced on Earth, with the tools available to us, over the last 10,000 years. How many different types of situations have we really been in, compared to the age and size of the universe.There are a couple problems here.
First, we don't know how big the universe of mathematically malleable situations is. Consequently, we also have absolutely no idea how extensive our sample is. It could, in fact, be very extensive.
Second, there is no reason to suspect our sample, no matter its relative extent, is not representative of the whole, or non-random. All the mathematically malleable problems we have experienced have reinforced the conclusion mathematics is inherently tied to the nature of the universe. Without delving too deeply into "likely", how likely is it that our sample is so completely biased in one direction, unless nature itself is biased the same way?
My concern is that we can always find, after the fact, some excuse like "laminar air flow" to convince ourselves that the problem was us, rather than with math.I think that objection would carry more weight if we had to create post-hoc new mathematics to cover such problems.
However, we don't. No one had to learn any new math (which would be the same as creating new physical laws) because of bumblebees.
Which ties right in to those Wall Street rocket scientists. If we were to rewind the tape, say, 10 years, and more correctly abstract their models (i.e., there can be a secular fall in house prices), the same mathematics would have led to a result much more consistent with nature.
Can't blame math for abstraction failures.
David,
Mathematics is inherently NOT tied to reality and, in fact, many would say that's its beauty. There's more than story of a theoretical mathematician who was saddened when someone discovered a use for his work in the real world.
2+2=4 is true in mathematics and many mathematicians could care less whether that mathematical expression would happen to applicable to the real world anywhere or everywhere and they (the theoretical ones anyway) will generally not claim that it does.
"The mathematicians and their banks all assumed that their mathematical models were tied to some underlying reality; that the constructs "really" had the value that the models suggested."
This isn't exactly true. All models are based on a set of assumptions - things that are assumed to be either true or "close enough". For example, that a distribution of returns is "close enough" to log normal. That returns can be assumed to be independent even when it's well known that they are not always. And so forth.
It's the mistaken assumptions and the not quite "close enough" after alls that get you. But that's not mathematics, at least not in the sense of 2+2=4.
Global warming is nothing but a power grab and also has nothing to do with mathematics.
"Essentially, all models are wrong, but some are useful."
Box, George E. P.; Norman R. Draper (1987). Empirical Model-Building and Response Surfaces. Wiley. pp. p. 424
Bret:
2+2=4 is true in mathematics and many mathematicians could care less whether that mathematical expression would happen to apply to the real world anywhereI don't follow.
Ellipses exist in nature. Mathematics can precisely describe any ellipse. That conjunction is not possible unless there is an inherent, invariant tie between mathematics and ellipses.
Turn the question on its head. Is there any part of nature that is not inherently tied to mathematics? (the wishes of mathematicians are irrelevant here)
Anonymous:
More accurately, although less memorably, all models are incomplete. Sometimes that does not stop them being useful.
I gave a presentation at a conference last week and a member of the audience objected that my model was incomplete.
I thought, but restrained myself from saying, that until I get around to explaining the Big Bang, all of my models are going to be incomplete.
Hey Skipper wrote: "Ellipses exist in nature."
Do perfect ellipses exist in nature? Perfect, unchanging, eternal? If not, mathematics can't really describe them. The best you can do is a time-varying statistical representation, but you have to make assumptions about the distribution of deviations from the perfect ellipse.
Once math is applied to the real world, it's all terribly messy and unelegant. Often useful, sometimes catastrophically inaccurate.
Do perfect ellipses exist in nature? Perfect, unchanging, eternal? If not, mathematics can't really describe them.Why not?
You are, in effect, asserting that there are influences upon an ellipse in nature that themselves are beyond mathematics.
I can't think of any offhand.
The earth travels about the sun in an ellipse - more or less. Since every particle in the universe affects the path, it's not a perfect ellipse and it is impossible to perfectly model its non-perfectness.
David, have you read Eric Temple Bell's 'Mathematics: Queen and Servant of Science'?
My physics adviser recommends it highly. Bell (you may know him as a science fiction writer) was his math teacher at Cal Tech.
I notice that the reviews at Amazon get Bell's proposition on its head.
'Queen' is the key word. As I understand it, Bell contends that the unreasonable usefulness of mathematics is an artifact. There are many times more mathematical theorems that have never yet found a congruence with a material system than there are that have been.
I guess another way would be to say that math exists outside the material systems but in some cases matches it.
2+2=4, so far as we can see, even in any of those alternative universes whose physical laws are unlike ours.
Assuming, I guess, that any universe worthy of the name will be self-coherent and not merely 'chaos and dark night.'
To put it in yet another way -- this time one you will not like -- it has been said that we can make one certain prediction about any universe that has life of any sort, no matter how unlike ours: It will transform through time according to the principles and Darwinian selection.
Bret:
I'm attempting to answer David's question: Is mathematics inherently tied to the nature of the universe, or is it just a really good modeling tool?Mathematics is a good modeling tool because it is inherently tied to the nature of the universe.
You are right, it is not possible to model the Earth's elliptical orbit exactly -- I noted that above. However, the fact that we do not have to create ad hoc mathematics to model various natural phenomena seems good reason to conclude mathematics and the universe are inherently tied.
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