Weeds will always try to find a way, but even without a geotextile weed membrane, if you lay slate chippings deep enough very few weeds are likely to grow.
One advantage of laying slate chippings without a weed membrane is that bulbs such as hyacinths, daffodils, crocuses and tulips will normally be able to grow through the slate while the vast majority of weeds will struggle to do so.