# Importance of rigor

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.

• I am pretty sure the harmonic series was believed to be convergent for a long time. There is even some funny quote on that but I can't find it anymore. May 1, 2012 at 16:21
• The divergence of the harmonic series was proved by Oresme, about $300$ years before Newton was born. I have read the argument, the idea is the same as the one we usually give to students. Bernoulli's argument was more manipulational. The proof by Oresme is buried in a philosophical/theological work, and had zero impact on mathematics. May 1, 2012 at 16:32
• @SimonMarkett If you've ever taught calculus, you know that the harmonic series is still believed (by students) to be convergent. May 1, 2012 at 16:42
• Basically, rigor marks the frontier between what might or might not be a good intuitive thought. Euler as some say, had a "genius intuition" so he never crossed the line, but most of us need that dose of rigor to guide us to correct ideas, and not slip into nonsense.
– Pedro
May 1, 2012 at 17:09
• @Simon: Godel only showed that CH is consistent with ZF (as is the axiom of choice). Only Cohen proved the undecidability of both from ZF. May 1, 2012 at 17:42

This was one of the top voted questions on Math Overflow. You'll find many interesting answers there.

Added: One of my favorite examples for regarding the unreliability of numerics is looking at the difference $\operatorname{li}(x)-\pi(x)$ where $\pi(x)=\sum_{p\leq x}1$ is the prime counting function, and $\text{li}(x)=\int_{2}^x \frac{1}{\log t}dt$ is the logarithmic integral. The prime number theorem tells us that $$\lim_{x\rightarrow \infty}\frac{\pi(x)}{\operatorname{li}(x)}=1,$$ but we can still ask what does $\operatorname{li}(x)-\pi(x)$ look like? If we create a plot going all the way to $10^{22}$, which is very far, it appears that $\operatorname{li}(x)-\pi(x)$ is always about $\sqrt{x}$. That is, it is increasing to infinity, and the data seem to suggest that we could find a monotonic function $f(x)$ about the size of $\sqrt{x}$ where $\operatorname{li}(x)-f(x)$ would be an even better approximation to $\pi(x)$.

However, this is not true at all. Littlewood proved that $\text{li(x)}-\pi(x)$ switches sign infinitely often, and we believe that the first time it does so is around $10^{316}$.

I first learned about this while reading the expository paper Prime Number Races by Andrew Granville and Greg Martin. I recommend reading it; it is accessible and interesting. The result and numerics I mentioned above are on pages 5 and 6. (At the start of section 2.)

• John Derbyshire mentions the same example in his book "Prime Obsession" about the Riemann Hypothesis. May 1, 2012 at 18:08
• Eric, could you provide a link to the graph of $\operatorname{li}(x)-\pi(x)$?
– Pedro
May 1, 2012 at 21:28
• @Look at page 6 of the paper Prime Number Races. There is a convincing chart with the values at $10^i$ for $8\leq i \leq 22$. May 2, 2012 at 8:40
• It is Littlewood, not Landau, who proved that $\mathrm{li}(x) - \pi(x)$ changes sign infinitely often. The first one to give an upper bound on the range of $x$ when this first happens is Skewes. May 2, 2012 at 16:16
• @FSica: Thank you for the correction, it was indeed Littlewood. Letting $x_1$ denote the first sign switch, as you mention the first upper bound was due to Skewes. He showed that $$x_1<10^{10^{10^{10^{34}}}},$$ the so called Skewes number. This was proven in 1933 conditionally on the Riemann hypothesis. Skewes later gave a new larger bound which was not dependent on RH, and several improvements over the years have led to Bays and Hudson's 1999 result which shows that $$x_1<1.4\times 10^{316}.$$ May 2, 2012 at 18:24

As your friends are non-mathematical, you might try examples they can understand--simple paradoxes. For example if the base and height of the red triangle is 1, then the length of each staircase below is two, so we have a simplistic argument that 2 = $\sqrt{2}$. This error, from a lack of rigor, would cause major troubles with Google maps, measuring to build a building ... (trying to stay non-mathematical here). There are many such easy paradoxes. (I know this argument is wrong--the points is lack of rigor can be dangerous)

There are times at which such naive dissection do give the correct answer, such as when we dissect a circle of radius $r$ into arcs and rearrange them into an approximate rectangle of (limiting) height $r$ and width circumference/2 = $\pi r$. Without rigor, we can not know when to trust our 'reasoning'.

• Maybe it be interesting to include some basic geometrical insight that $ds \neq dx+dy$ but rather "$ds^2 =dx^2+dy^2$" which is not rigorous but rather intuitive.
– Pedro
May 1, 2012 at 17:07
• @PeterTamaroff it might, but then you are bringing in the missing rigor, I thought the point was to show missing rigor causes big problems! My suggestion is just do it on a third grade level instead of using one of the other very impressive answers here that may not convince the non-mathematical. May 1, 2012 at 17:09
• @Chris I see. ------------
– Pedro
May 1, 2012 at 17:12
• This example can be usefully combined with that proof that the area of a circle is $\pi r^2$ which cuts the circle up into wedges and rearranges them to form an approximate parallelogram. That argument asserts that the limiting rectangle is $r\times \pi r$, but the naive argument for these dimensions assumes that arclength is preserved under limits, which the above example shows is not the case in general. So why is this idea right in one argument but wrong in the other? Getting to the bottom of this requires getting more rigorous.
– user21467
May 1, 2012 at 17:55

The "era of rigor" in mathematics began about 150 years ago, a time in which a lot of the mathematics before this era was turned rigorous and cleared from many mistakes. For this reason it is not very trivial to come up with good examples without knowing a lot about mathematical history.

One of the reasons that pre-rigorous mathematics is still useful is that it was essentially created to help developing physical theories. There was no real theoretical mathematics as we have today, and in fact if you look closely you will see that many mathematicians from that time were also physicists, chemists and alchemists (not to mention dark wizards and theological nuts).

I can give a similar example about the axiom of choice, which as a mathematically inclined person you can probably relate to. Consider the mathematicians of exactly a century ago. Many refused to accept the axiom of choice and it was not fully clear whether or not it is consistent or preposterous. Only after Gödel's work people began relaxing the constant questioning (although you still see people today which claim that the axiom of choice is false).

Compare the above situation to the common mathematician of 2012. The vast majority of mathematicians just assume the axiom of choice so bluntly that they use it for things which it is not even close to being needed for. In fact I hear set theorists who admit they have absolutely no intuition whatsoever how things work without the axiom of choice. Of course this can be remedied, but people will not really put that much effort into because they prefer to use their time for better things.

The case is similar to the importance of rigor because we are so deep into the age of rigor that we have essentially forgotten how non-rigorous mathematics look like. You can see it under the surface when talking to a mathematician on something that he is currently working on, but even then people are usually tend to be careful with their words as to avoid claiming false claims.

The most important use of rigor, I think, comes from logic and set theory in the form of the various paradoxes of set theory. When set theory was first developed by Cantor there was still not enough rigor in mathematics in order to well-define all the notions needed (Cantor's paradox is an exceptionally good example). One excellent example is König's theorem. König figured out a contradiction in set theory but it was a mistake in his understanding originating in a not-so-rigorous proof of some claim that he read. The result is actually one of the basic theorems in cardinal arithmetics.

It is important to understand that once we began dealing with the transfinite it was no longer "intuitive" to our physical intuition how things should work. This caused many people reject these ideas and belittle them (transfinite related ideas, the axiom of choice, and so on). Nowadays we know that if one does not use intuition coming from the physical world, but rather develops a mathematical intuition based on the definitions and the axioms of mathematics - one can have a good grasp how the transfinite basics work.

Other good examples are the Weierstrass function which is continuous but non-differentiable anywhere. Before that it was commonly believed that all continuous functions were differentiable almost everywhere. Gauss himself used topological arguments to prove the fundamental theorem of algebra, although he did not prove these arguments.

All in all, in mathematics we have "notion" and "definition". Rarely these two concepts coincide but usually we think about objects in one way and we define them to be approximately what we would like them to be. The introduction of rigor made sure that we work by the definition, and step by step while being guided by the notion - and not the other way around.

• "There was no real theoretical mathematics as we have today, and in fact if you look closely you will see that many mathematicians from that time were also physicists, chemists and alchemists." I would like to read more about this idea, do you know any good sources ? Jan 25, 2015 at 1:30
• On which part? Generally it's just picking up biographies. Newton was a physicist and an alchemist. Fourier wasn't really a mathematician to begin with, and many others were simply scientists in the broad term of the word. As for rigor, just look at Euler, Cauchy, even as far as early 1900s you can find mistakes rooted in lack of correct rigor. Foe example König's theorem in set theory (not about trees) was an attempt to disprove the axiom of choice, but it was just due to formal misunderstanding (or rather the ambiguity) of definitions in set theory at the time. Jan 25, 2015 at 1:38

Cauchy and Fourier had a famous dispute about whether Fourier's use of trigonometric series to solve the heat equation was valid. To a large extent, the basic rigorous notions of analysis were formalized in order to resolve this dispute. Along the way, Cauchy published a famous "wrong" theorem. (It wasn't really wrong; it just used a different definition than that which became standard.)

Here's another, more applied, example. I remember going to an joint meetings talk years ago in which the speaker described a situation in which engineers had employed widely-used software to numerically compute solutions to PDE's. But they had neglected to prove that, at the level of precision they were using, their methods would converge to the correct solution. The result was disastrous: a brand new 180 million dollar oil rig collapsed so thunderously that it registered a 3 on the Richter scale.

EDIT: Thanks to Kcd here's the link to more details.

• It was not multi-billion or 4.0... but close. Details on your second example are at failures.wikispaces.com/…. Here is a quote from the middle of that page: "The sinking was partially caused by the inappropriate use of finite element-code with regards to the global analysis of the finalized design: the finite element mesh used to analyze the tri-cells was too coarse to predict the shear stress accurately. The flawed analysis and post-processing of the tri-cell design led to shear forces being underestimated by some 45%. [contd.]"
– KCd
May 3, 2012 at 4:29
• "Another flaw with the use of the software was the human error involved: the design software involved considerable complexity, which led to a a very high perception of precision by those who used it. One of the features of this software was the way it ran calculations: it would only flag certain sections that were deemed critical by the software presets, and indicate those as sections that the engineers on the design team needed to check. The probable failure point of the SLA-1 GBS was not flagged."
– KCd
May 3, 2012 at 4:29
• @Kcd Thanks. I was trying to remember the details from a talk I was at years ago. Nice to have the reference for the correct details. May 3, 2012 at 4:49
• Another link for this story on the Sleipner platform accident is at www5.in.tum.de/~huckle/sleipner_Jakobsen.pdf.
– KCd
May 4, 2012 at 3:06

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.