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Comment by libeclipse | Hacker News Reader

> The Facebook researchers compared their method to only a few of Mathematica’s functions —“integrate” for integrals and “DSolve” for differential equations — but Mathematica users can access hundreds of other solving tools.

> [...] it only included equations with one variable, and only those based on elementary functions. “It was a thin slice of possible expressions,”

> The neural net wasn’t tested on messier functions often used in physics and finance, like error functions or Bessel functions. (The Facebook group said it could be, in future versions, with very simple modifications.)

> Other critics have noted that the Facebook group’s neural net doesn’t really understand the math; it’s more of an exceptional guesser.

> Still, they agree that the new approach will prove useful.

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> exceptional guesser

Considering neural networks are inherently maximizing probabilities and statistical descriptions of data, this should come as no surprise. This work has not dissolved the dichotomy between rules-based and statistical methods, but rather transmuted the syntax of rules-based expressions into a representation that can be exploited by statistical machines in a way that makes "guessing" more fruitful.

There are some examples near the end of the paper showing how the authors take an initially intractable expression and are able to simplify it with their approach so that Mathematica can actually perform the integral for them. It seems much more appropriate to market this method as a preprocessor for massive expressions to a more chewable size.

> exceptional guesser

This IMO describes how mathematics itself moves forward... A matematician is an extremely well-trained 'guesser' who is also able to sink a lot of time into formal verification.

The process is essentially: a) Find an interesting conjecture that you've got a strong guess to be true. b) Check for obvious (or less obvious) counterexamples, or conflicting theorems. c) Prove the thing is true.

A large part of the art of being a working mathematician is in part (a): you need to make a really good guess. An ideal conjecture is correct AND proveable AND leads to other interesting results, or says interesting things about bigger problems.

So what happens when we apply really good versions of current AI to this area? Picking out an 'interesting' conjecture is still Strong-AI-Complete: it requires lots of domain knowledge, and an understanding of what this particular conjecture would 'unlock.' But we could perhaps come up with good 'guessers' which quickly tell us whether a given idea might work out, perhaps saving a bunch of effort. Perhaps we could even get to the point of generating a proposed proof which can be fed to an automated proof checking system, allowing for inspection and modification by the human in the loop.

I like this prospect:

> Another unsolved problem where this approach shows promise is one of the most disturbing aspects of neural nets: No one really understands how they work. Training bits enter at one end and prediction bits emerge from the other, but what happens in between — the exact process that makes neural nets into such good guessers — remains a critical open question.

> Symbolic math, on the other hand, is decidedly less mysterious. “We know how math works,” said Charton. “By using specific math problems as a test to see where machines succeed and where they fail, we can learn how neural nets work.”

> “By using specific math problems as a test to see where machines succeed and where they fail, we can learn how neural nets work.”

XOR and Spiral benchmark was studied in neural networks since 70's.

The criticisms read as good starting points for subsequent research.

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