TITLE: Importance of rigor
QUESTION [29 upvotes]: I always have a hard time explaining the importance of rigor to my friends who are not mathematically minded. A lot of past mathematicians develop the foundations of today's mathematics without going through all the rigor but their work is still relevant now, e.g. Newton and Leibniz's calculus and Euler's work on infinite series.
Is there a historical event where a wrong conclusion is derived due to lack of rigor and cause a harmful effect?
Edit: Thanks so much for the lively discussion! I just want to point out that I'm not interested in counterintuitive results or paradoxes, instead I'm looking for instances where wrongly derived conclusion caused a disastrous effect.

REPLY [3 votes]: It seems that all the proposed answers so far have been about why math might get wrong answers without rigor. While that's been shown to be a nontrivial risk, I would think non-mathematicians might be interested in how rigor gives math a unique position as a human activity. 
The basic point would be that 18$^{th}$ century and older writers in all kinds of genres used to throw around the phrase "I shall prove..." about all sorts of notions that the mathematically inclined would never admit as a proof today. Yet at the time, there was absolutely nothing on the scene that worked better. Euler's results were and are astounding but, as they say, to derive Euler's results with Euler's methods one has to actually be Euler.
Since the gift of rigor to mathematics by Cauchy, Weierstrass, Cantor, Hilbert, Noether, and others, we've found ourselves in a field that seems completely unique among human activities in this: suppose the mathematical community disagrees about the validity of some new proposed proof of a mathematical proposition. It's an incredibly safe bet that, come back five years later, and the experts in the field will now be in agreement. That's something essentially nobody else has, though of course certain sciences have a closer or further approximation of it.
I should leave a caveat that none of this applies to opinions, about, say, what math is worthwhile-lots of people objected to set theory in 1900, Bourbaki in 1950, and category theory today. It's about specific proofs of well-defined facts that mathematicians don't have to agree to disagree.