What does 'convicting purpose' mean?
Educated-class anxiety can often be characterized as a feeling overabundant of options without a core of convicting purpose. It’s worth noting that rich countries are more anxious than poorer ones.

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I agree with Michael that 'a feeling overabundant of options without a core of convicting purpose' doesn't make any sense.

All I can imagine is that the NY Times journalist responsible for this posted it without checking what he'd written.

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I've never seen this expression in my life and it is in a sentence which contains some nonsense as well ("a feeling overabundant of options").  I recommend that you ignore this text as it will not help you learn English.

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I'm American, and I agree with SuKi and Michael.  The article is not written in proper English. Here's a better newspaper:. 
http://m.washingtontimes.com