Statements which were given as axioms, which later turned out to be false.

Question

I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom. (For example, there is no completeness axiom in Cauchy's Cours d'Analyse).

I also know that mathematicians in the early nineteenth century had some false beliefs about real analysis which were later uprooted as the process of rigorization continued. For example, it was widely believed that a continuous real function must be differentiable except at isolated points - a claim which Weierstrass refuted in the 1870s by defining a function that is continuous everywhere but differentiable nowhere. (http://en.wikipedia.org/wiki/Weierstrass_function).

What I'd like to know is whether at any point an "axiom" was proposed for real arithmetic, which subsequently turned out to be false.

I'd also like to hear about such cases from other branches of mathematics.

However, I'm not so interested in cases of inconsistent sets of axioms (e.g. Gottlob Frege's Grundgesetze, or Church's first formulation of the lambda calculus).

The best example I've found so far is Leibniz's "principle of continuity", according to which "what is true up to the limit is true at the limit". Apparently this was sometimes called an "axiom" although it is obviously not true in general. (I'm getting this from Lakatos's Proofs and Refutations, pg. 128). I'm not entirely happy with this example, because it's so obviously false that I can't believe that it was accepted, except as a heuristic.

Thanks in advance!

Answer 1

Another example from real analysis would be the question of the pointwise convergence of the Fourier series of a continuous function (defined on a closed interval). Many people, including Dirichlet and even the master rigorist Weierstrass himself, believed that the Fourier series of such a function converges pointwise everywhere to the function itself. Some clung on to this belief so strongly that they even viewed it as an infallible axiom.

Hence, one can imagine the great upset when, in 1876, Paul du Bois-Reymond proved the existence of a continuous function whose Fourier series diverges at a point. His proof is non-constructive and uses a method called the principle of condensation of singularities. I have absolutely no idea how the method works, but I do know of a very common proof that uses the Baire Category Theorem (using the Baire Category Theorem, one can also prove the existence of continuous functions that are not differentiable at any point).

After the dust had settled in the wake of du Bois-Reymond's seismic discovery, people started fervently believing that there should exist a continuous function whose Fourier series diverges everywhere - an opinion that lay on the other extreme! Andrei Kolmogorov inadvertently lent support to this claim by exhibiting, in 1926, an $ {L^{1}}([- \pi,\pi]) $-function whose Fourier series diverges everywhere. However, there was great upheaval once more in Fourier-land when the combined efforts of Lennart Carleson and Richard Hunt in the late 1960's showed that the Fourier series of any $ f \in {L^{p}}([- \pi,\pi]) $ converges almost everywhere to $ f $, for all $ p > 1 $ (this result subsumes the case of continuous functions). During an interview with the AMS, Carleson revealed that he had originally tried to disprove his result (pertaining to $ p = 2 $), but in the end, his failure to produce a counterexample convinced him that he should be working in the other direction instead.

Therefore, in the field of Fourier analysis, viewpoints have changed and cherished beliefs have been destroyed - twice.

Answer 2

One of the most well-known examples is Frege's axiom of general comprehension, which asserts that for any definite property $P$, one may form the set $$\{x\mid\ P(x)\ \},$$ consisting of all objects $x$ with property $P$. This natural-seeming principle is one of the main axioms of what is now known as naive set theory, and formed a central axiom in Frege's Begriffsschrift and later the Grundgezetze (which I see now that you said you weren't interested in, oh well), intended as a formal logical foundation of arithmetic and all mathematics. But the axiom was famously refuted by Betrand Russell with the Russell paradox, showing that there can be no set $R=\{x\mid\ x\notin x\ \}$, consisting of the sets $x$ that are not members of themselves, since then $R\in R\iff R\notin R$, a contradiction.

Answer 3

In the mathematical theory of social welfare, it is possible to create a list of axioms that lead to a contradiction. For example, in voting theory, the following axioms for a voting system are considered reasonable in order for the system to qualify as being fair:

1. Each voter can have any set of rational preferences. This requirement is called “universal admissibility”.

2. If a voter prefers Candidate A to Candidate B, and Candidate B to Candidate C, then he/she prefers A to C. This requirement is called “transitivity”.

3. If every voter prefers A to B, then the group prefers A to B. This is sometimes called the “unanimity” condition.

4. If every voter prefers A to B, then any change in preferences that does not affect this relationship must not affect the group preference for A over B. For example, if a set of historians unanimously decides that Abraham Lincoln was a better president than Chester A. Arthur, a changing opinion of Bill Clinton should not affect this decision. This more subtle requirement is called “independence from irrelevant alternatives”.

5. There are no dictators. In other words, no voter exists whose preferences determine the preferences of the whole group.

The mathematical economist Kenneth Arrow showed in a landmark paper (stemming from his PhD thesis) that one obtains a contradiction if all five assumptions are assumed to hold. In fact, Assumptions (1) - (4) imply the existence of a dictator. However, these assumptions seem fairly reasonable and consistent, so the fact that they are contradictory is why Arrow named his paper “A Difficulty in the Concept of Social Welfare”. His result is known nowadays as Arrow's Impossibility Theorem.

Answer 4

Perhaps one of the earliest examples would be with the Pythagoreans, who held that any two magnitudes were commensurable, measured as integer multiples of a smaller common unit, a belief that was connected with their mystical religious views and also with their mathematical theory of musical harmony. The Pythagoreans were shocked by the discovery of incommensurable numbers, such as $\sqrt{2}$.

But it may be anachronistic to refer to the fundamental Pythagorean beliefs or principles as "axioms".