One of my friends was asking me about tertiary level mathematics as opposed to high school mathematics, and naturally the topic of rigour came up.
To provide him with a brief glimpse as to the difference, I said the following.
In high school, you were taught that the area of a rectangle is ab
where a is the breadth and b is the height. You can physically see
this by constructing an a×b grid and counting the squares it
forms, provided a and b are integers.
He had agreed and said that it was “obvious” that the area of a rectangle was ab. I then responded with:
What is the area of a rectangle with dimensions π by √2?
He immediately just said π√2, and then I responded with one of the most common questions in mathematics:
How do you know that for sure?
I had said that it intuitively works for integer values of a and b, but how do we KNOW for sure that it works for irrational values of a and b? Then I used that as a gateway to explain that in tertiary level mathematics we don’t assume such things. There is no “It is clearly true for these easy-to-understand integers, so therefore it is true for all real values” and that everything must be proven.
He then asked me something that I had no answer to:
I get that we cannot assume these kinds of things, but has there ever been an occasion where an assumption or a lack of rigour has killed someone before?
I am sure that there may exist an example floating somewhere in history, but I cannot think of any.
Do you know of one?
EDIT: Cheers to starsplusplus
A lot of really great responses! However, the majority of them don’t quite fit the definition of ‘rigour’ in the mathematical sense, which is vastly different to the common English term. See this. Many of the answers provided so far have been accidents/deaths caused by a lack of what I feel to be more like procedural rigour as opposed to mathematical rigour.
It seems like further clarification is needed regarding what I’m looking for in a response. I was looking for an example of where an individual(s) did something that was mathematically incorrect (not a trivial computational error though) that had a consequence which led to the death of one or more people. So, what do I mean by something mathematically incorrect that isn’t a trivial computational error?
Example of something I’m looking for:
Say somebody is part of programming or working out the math behind a missile firing mechanism. In part of their computations, they did one of the following which yielded an incorrect value. This incorrect value caused the missile to fly out of control and cause the death of one or more person(s).
- Exchanged a summation with an integral unjustifiably
- Needed to use two sequence of numbers that always yielded relatively prime numbers. They used a computer but it didn’t find any counter examples, so the programmer assumed that the formula always yields relatively prime integers. However, the counter example lies at n=99999999999999999999999999, beyond reasonable computational time.
- The limit of a series was to be used at some point in the computations. To calculate it, the person re-arranged terms however they liked and then found a limit. But the series didn’t converge absolutely so they could have gotten any value.
Particularly, lack of knowledge of Bayes’ theorem, and intuitive use of probability, lead to many misdiagnosed patients all of the time. In fact, some studies suggest that as many as 85%(!) of medical professionals get these type of questions wrong.
A famous example is the following. Given that:
- 1% of women have breast cancer.
- 80% of mammograms detect breast cancer when it is there.
- 10% of mammograms detect breast cancer when it’s not there.
Now say that a woman is diagnosed with breast cancer using a mammogram. What are the chances she actually has cancer?
Ask your friends (including medical students) what their intuition regarding the the answer is, and I’m willing to bet most will say around 80%. The mathematical reasoning people give for this answer is simple: Since the test is right 80% of the time, and the test was positive, the patient has a 80% chance of being sick. Sound correct?
Would you be surprised to learn that the actual percentage is closer to 10%?
This perhaps surprising result is a consequence of Bayes’ theorem: The overall probability of the event (breast cancer in all women) has a crucial role in determining the conditional probability (breast cancer given a mammography).
I hope it’s obvious why such a misdiagnoses can be fatal, especially if treatment increases the risk of other forms of cancer, or in reversed scenarios where patients are not given care when tests give negative results.