I think Wolfram Alpha is an elaborate advertisement for Mathematica.

No, it doesn't compete with search engines, but it will fail for the same reason that Microsoft search failed as explained above.

If it's not a search engine, then what is the significance of it failing as a search engine?

But your suggestion of WA as an ad makes sense - but what is held back in this freemium hypothesis, that will make users need Mathematica?

As for why people would buy Mathematica: companies are not going to give away their trade secrets over the web via queries made to Wolfram Alpha.

That's a compelling reason for some users. I thought WA would be slow, and that would be a reason to buy a local version (or a subscription to a faster version), but it surprised me how well it coped with the launch load. But after seeing your freemium idea, I've changed my mind to think they'll just leave off some features - that's the standard approach.

The notion of having every query answered in a reasonable way applies to all sorts of things -- not just search engines.

I suspect that Microsoft's failure in this regard would play out in many other contexts including this one. Imagine for example if Mathematica were to have a much higher failure rate.

Mathematica works as well as it does because an enormous amount of research has been put into one domain. Wolfram Alpha is not like that at all.

No one expects Mathematica to be a search engine. However, many people (not just you) expect WA to be a search engine, even though it clearly is not one.

It's a matter of expectation.

And it seems likely that creating this expectation is a deliberate ploy on WA's part to create publicity. "Google killer" is a much better media story than "mathematica online". They've designed the front page in a google-style, with a similar text interface, and probably many other ways I haven't noticed.

yup, why compete when you can get acquired.