Math is but a medium for modeling. Any model can have predictive power.

The challenge is in determining which surprising aspects of the model have faithful correspondence to the domain and which are nonessential artifacts of the modeling medium.

No.

The surprising thing is that the models generate predictions far beyond the domains they were designed for (and far beyond the original knowledge of the people making the models), and that the predictions are so mindbogglingly accurate that there seems to be Something Else going on.

See the Unreasonable Effectiveness of Mathematics link below.

> The surprising thing is that the models generate predictions far beyond the domains they were designed for (and far beyond the original knowledge of the people making the models), and that the predictions are so mindbogglingly accurate that there seems to be Something Else going on.

Yes -- and I can't resist citing antimatter as a perfect example. Dirac's Equation had two symmetrical results (sort of like a quadratic), and Dirac wasn't sure about the physical meaning of one of them. After antimatter was detected, Dirac was asked why he didn't simply predict antimatter himself based on a literal interpretation of his equation's implications. He replied, "Pure cowardice."

But in the end antimatter didn't have negative energy anyway?

Yes, true, the term "antimatter" has always been somewhat of a misnomer. "Antimatter" differs from matter in a number of ways, but there's only one kind of gravitational mass-energy.

No?

Surprising thing: Math's not the only modeling medium that can be Unreasonably Effective.

Unreasonable Effectiveness of Mathematics: Know; saw; upvoted apropos reference already.

And the point is that it's not about modeling!

What happens is that we invent crazy math that is not supposed to have applicability, then some years go by and it's like, oh my God, quantum mechanics is somehow exactly all about the operation of unit Hermitian matrices... how crazy is that?? etc.

If it were some kind of model that we are able to successively refine, the progress of discovery would look something like a Taylor series, and it would be no surprise that we are eventually able to model phenomena within some tolerance epsilon.

But that is not what is happening! Rather, it's that we discover that some large and sophisticated piece of math, for which we had not thought of any particular applicability, turns out to exactly represent specific advanced physical phenomena. This happens over and over.

I'm confused by your statement that "Math's not the only modeling medium that can be Unreasonably Effective", because it's not clear to me what the notion of a "non-mathematical model" means. (That is to say, it's not clear to me that you're even correct in the weaker assertion that "Math's not the only modeling medium.") Can you explain?

Modeling media besides math: digital electronic circuits, analog electronic circuits, legos, quantum mechanical phenomena, gears, computer simulation, wetware, mythology, clocks, BZ reactions, ...

Several of those are, in fact, math. (In particular, computer simulation.) The others are mostly phsyical models, which I must admit, I had not thought of. So that yields two; mathematical models (that is to say, formal models) and physical models (that is to say, real-world models). (The line can perhaps be a bit blurry in the case of e.g. digital electronic circuits, but, well, I'm not claiming there's a sharp line.)

That leaves "mythology", which I'm not convinced is a proper modeling medium at all.

But those are examples of modeling math, with a superficial layer of physical material. Another way to say it is the degree that these are useful models, is the same degree to which they reflect mathematics. With the possible exception of mythology.

Many of which are reducible to math. Though I'm not sure what legos have been "unreasonably effective" at predicting.

Did you read the essay? Actually, both of them? The reason mathematics is considered Unreasonably Effective is because we have never seen anything with similar effectiveness. Ever.

Since math is just the formalization used to describe quantitative patterns observed, you're already seeing math as purely as it will ever get. Really, it's the other way around--since math is an abstraction of observation, what you observe with your senses day to day is a much more direct and fundamental reality. Math is a language, and reality is what it talks about. Similarly, people who speak English will find that the most effective visualization and perception of what they speak about is to just directly see and perceive the thing that they talk about.

That's why I've never thought that there's any "Unreasonable Effectiveness of Mathematics in Describing the World". Math is derived from reality and describes it insofar as it can be observed to be given a description, and so it's unsurprising that math is effective at describing what it's designed to effectively describe. It's like saying that English is unreasonably effective at talking about things.

The surprising bit is not that it explains the observations that it was modelled on, but that it explains further, unexpected observations that were unknown at that time.

It's not that maths is so great, or even that models are great, but that we were able to model some essence of a phenomenon, beyond what we witnessed.

This happens not because it's a model (you can have as many models/theories as you like), but because we selected that model, usually due to beauty/parsimony/elegance - which amounts to some version of Occam's Razor: to not make a model more complex than needed to explain the observations. (do not multiply elements unnecessarily; given different models that explain the observations, the simplest one is most likely the truest of them). http://en.wikipedia.org/wiki/Occam%27s_razor

The real puzzle (if it's a puzzle) is why Occam's Razor works so well...