Consequences of lack of rigour [closed]

Question

The standards of rigour in mathematics have increased several times during history. In the process some statements, previously considered correct where refuted. I wonder if these wrong statements were "applied" anywhere before (or after) refutation to some harmful effect. For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?

Sorry if this is a silly question.

Edit: Let me clarify. I am looking for examples of bad things happening, which fall in the following scheme:

1. Lack of rigour led to a wrong statement (by "today's standard") in pure mathematics.

2. That wrong statement was correctly applied to some other field, or somehow used in calculations etc.

3. The conclusion from item 2 was then applied to a real-world situation, perhaps in construction or engineering.

4. This led to some real-world harm or danger.

It is crucial that there was no mistakes or omissions in 2,3,4, and hypothetical being omniscient on the level of 1 would approve the project.

Sorry if I made it even sillier. Feel free to close the question.

Answer 1

Quoting from

Martin Gardner: Curves of constant width, one of which makes it possible to drill square holes, Scientific American Vol. 208, No. 2 (February 1963), pp. 148-158:

Is the circle the only closed curve of constant width? Most people would say yes, thus providing a sterling example of how far one's mathematical intuition can go astray. Actually there is an infinity of such curves. Any of them can be cross section of a roller that will roll a platform as smoothly as a circular cylinder! The failure to recognize such curves can have and has had disastrous consequences in industry. To give an example, it might be thought that the cylindrical hull of a half-built submarine could be tested for circularity by just measuring maximum widths in all directions. As will soon be made clear, such a hull can be monstruously lopsided and still pass such a test. It is precisely for this reason that the circularity of a submarine hull is always tested by applying curved templates.

Answer 2

In classical mechanics, dissipative forces are typically regarded as having a stabilizing effect. However, this is not always the case as the folks behind the first satellite launched by the United States Explorer I found out. To quote from the linked wikipedia article

Explorer 1 changed rotation axis after launch. The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements. Later it was understood that on general grounds, the body ends up in the spin state that minimizes the kinetic rotational energy for a fixed angular momentum (this being the maximal-inertia axis). This motivated the first further development of the Eulerian theory of rigid body dynamics after nearly 200 years—to address this kind of momentum-preserving energy dissipation.

In short, the satellite ended up rotating like a windmill blade because of a counterintuitive phenomenon known as dissipation-induced instabilities; for a review article on this see

Krechetnikov, R.; Marsden, J. E., Dissipation-induced instabilities in finite dimensions, Rev. Mod. Phys. 79, No. 2, 519-553 (2007). ZBL1205.70002.

Answer 3

This is a riff of my answer to a similar question on Math SE. It seems to better fit to this question anyway.

I mainly argue that it is highly unlikely that you will find such an example. To this purpose, let’s look at your steps and what they would imply in practical situations:

1. Lack of rigour led to a wrong statement (by "today's standard") in pure mathematics.

Wrong statements from pure mathematics are made for a reason, namely that they hold in most cases the mathematician in question (and their peers, etc.) can think of. Since mathematicians are usually good of thinking about relevant cases, this means that the cases in which such a statement fails are either relatively few or standing out from examples inspired by application. This alone makes it unlikely that such a case matters in application. And that’s not even speaking of the several cases where the counter-example does not matter in reality at all.

In cases like “every continuous function is ‘mostly’ differentiable” this may be less obvious. I address this separately below.

2. That wrong statement was correctly applied to some other field, or somehow used in calculations etc.

Good science doesn’t just apply mathematics in a fire-and-forget sense. It involves a lot of practical tests, double-checking, simulations, checking known cases, thinking about mathematical conditions and limitations, etc. This is usually not because the underlying mathematics may be wrong, but because it may be applied for the wrong reason, etc. Either way, these safety checks should also find problem due to wrong underlying mathematics. Again, this may fail to find the error, but it’s unlikely.

Moreover, in a considerable amount of cases, the application may happen the other way around: A scientists makes an empirical observation and then searches for the mathematics that allows to prove it. Now, to allow for a (true) empirical observation, the mathematical statement must hold in a considerable number of relevant cases, which makes it unlikely to fail.

Of course, a lot of things may go wrong in this case, but then it’s a case of wrongly applied mathematics, lack of scientific rigour, etc.

3. The conclusion from item 2 was then applied to a real-world situation, perhaps in construction or engineering.

This is mostly like step 2: Engineering again involves a lot of testing, etc. that should detect the problem.

Conclusion

So, for lack of rigour actually causing some real-life damage, a lot of unlikely things must coincide. You can somewhat reduce the number of required coincidences by looking at one of those few cases, where mathematics is applied very directly in a case which does not allow for extensive testing (say space travel) though.

Appendix: What about fractals?

Now you may say: “Fractals are ubiquitous in nature and a counter-example to every continuous curve being ‘mostly’ differentiable”. (This other answer is based on this.) From this you may conclude that my argument that counter-examples to wrong statements are sparse in reality does not hold.

However, applying the entire notion of continuity or differentiability to non-atomic objects is already artificial in a way that should be clear to everybody applying mathematics. Real fractures, coast lines, romanesco etc. are made of grains, cells, atoms, and similar, which in turn ultimately are not continuous in a way that translates to the macroscopic object. Mind that this is not about the question whether reality is continuous or differentiable, but about how this manifests or can be applied. For example, if you work with a derivative, you do not care about it for its own sake, but a physical meaning connected to it. This physical meaning does not persist all the way down. If you work on the assumption that your object has a physically meaningful derivative at these scales, you are making a scientific mistake anyway.

From another perspective, there are no fractals in the mathematical sense in reality. (Fractals are “only” a very good way of modelling certain natural phenomena, the same way that continuity, derivatives, etc. are.) For every realistic approximation of a non-differentiable curve in a macroscopic object, you will find an equivalent approximation of a differentiable curve.

So, to summarise, if somebody is really building something based on the assumption that continuous curves are “mostly” differentiable, they make at least one other, scientific mistake anyway.

Answer 4

For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?

Probably

The paper "Fracture and Damage Behaviors of Concrete in the Fractal Space" compares the fracture toughness of concrete when you take fractals into account v.s. when you do not take them into account. Their experiments suggest that the fractal analysis is more accurate, and that the non-fractal approach often overestimates the strength of concrete (which is obviously bad for bridges).

Functions whose graphs are fractals just so happen to be examples of continuous functions that are no where differentiable. In fact, the first known example of such a function had a fractal graph. Therefore, it is plausible that a bridge engineer, assuming no such functions exist, overestimated the strength of concrete, which concrete was the cause of the bridge's failure. However, it is unlikely that he thought to apply fractals in determining the cause of failure as well, so we may have trouble finding specific examples.

In general, fractals occur very commonly in nature. In many contexts, fractals are the only thing that occur (try finding a natural coast line that is differentiable). Therefore, I suspect that concrete is not the only time that this assumption misled us gravely.

Easier to prove, however, is the application that fractals, and the modern theory of functions, has given us. And it all started with a question: How weirdly can a function act? Being blinded by "intuition" can have us miss the forest for the well-behaved trees.