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I found this question that discusses abstract theories that later found application. I am interested in accepted (at least at one point in time) abstract theories that:

  • was contradicted by attempts to apply and/or observed phenomenon (as currently understood).
  • are irreconcilable with another accepted and intersecting abstract theory

I am a licensed engineer by profession; but turning to philosophy. Some individuals point to the a priori or abstract nature of mathematics as justification that other a priori forms of knowledge exist. I hope to demonstrate that this dependence is unreliable by showing that mathematics requires a posteriori verification to acquire meaning.

I would be happy with 2–3 examples, or if someone could refer any texts that might document the history of such errors that would be helpful.

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    $\begingroup$ Dear @Lugh: I think your question is a little 'muddled' in parts, which may explain the downvote. E.g., (0) in your title you write "abstract proofs" (which by itself is somewhat pleonastic or even non-existant, a usual collocation is "formal proof"), but then in your question you are writing about something else, namely "abstract theories". Theories are not the same as proofs. In mathematics, a theory is (the deductive closure of) a set of sentences. Usually, this is not called a proof. (1) I can't see what you mean by "this dependence"; (2) "verification to acquire meaning" [...] $\endgroup$ – Peter Heinig Sep 17 '17 at 8:22
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    $\begingroup$ "verification to acquire meaning" seems (to me) a rather unusual way of using the word "meaning". Meaning, often, is more-or-less a purely subjective concept, which cannot be 'verified'. $\endgroup$ – Peter Heinig Sep 17 '17 at 8:23
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    $\begingroup$ Only briefly: I think your overall project is reasonable and noble, yet impossible, at least modulo to what is by far the most usual philosophical view on this matter: a proof/abstract theory cannot be refuted by empirical data, more or less by definition of abstract proof. All you can do is deny the logic used in the proof in question, yet this is a willful decision, and often not a reasonable one, and many people will not agree with you that the logic you choose to deny even can be denied. Within reason, it's good that anyone is allowed to do so, yet you won't convince anyone. $\endgroup$ – Peter Heinig Sep 17 '17 at 8:55
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    $\begingroup$ A general piece of advice: it might be good for you and your project to pay attention to Immanuel Kant's philosophy and its reception. It seems too much of a statement to recommend to you even one source where to start (and maybe you even already have started on this), yet I think one should point out to you that what your undertaking is by no means a new one, it has been much discussed in the last three hundred years or so. This need not be discouraging: there are many good modern treatments, and even today you might find out something new. $\endgroup$ – Peter Heinig Sep 17 '17 at 10:06
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    $\begingroup$ I think a 'standard' example for what you seem to be asking for is Non-euclidean geometry. I won't have time to write you more of a summary than this one: for quite some time most people seem to have assumed that Euclid's theory was empirically true, which roughly I take to mean: 'Euclid - (parallel postulate)' has only one model in the empirical world, and in that model, the parallel postulate is true (and hence the parallel postulate is semantically-entailed by 'Euclid-(parallel postulate)'. This view was empirically refuted. [...] $\endgroup$ – Peter Heinig Sep 17 '17 at 12:09
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You can see Imre Lakatos' book: Proofs and Refutations, with a beautiful discussion of some case histories of theorems whose proofs had to be "revised" in order to take into account counterexamples.

See: I.Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge UP, or.ed.1976.

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I'm not sure if this is what you are interested in, but two supposed proofs of the four color theorem appeared in the XIXth century and were accepted as correct proofs for eleven years, but both turned out to be wrong. Quoting form Wikipedia:

One alleged proof was given by Alfred Kempe in 1879, which was widely acclaimed; another was given by Peter Guthrie Tait in 1880. It was not until 1890 that Kempe's proof was shown incorrect by Percy Heawood, and in 1891, Tait's proof was shown incorrect by Julius Petersen—each false proof stood unchallenged for 11 years.

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  • $\begingroup$ +1: I think this answer is very relevant, and really addresses (what seems to me the most reasonable interpretation of) the OP. All refutations of alleged proofs to the Four Color Theorem are more-or-less empirical/semantical refutations. Cf. e.g. (0) any good account of Kempe's proof, and (1) this thread. $\endgroup$ – Peter Heinig Sep 17 '17 at 10:20
  • $\begingroup$ @PeterHeinig Thank you. It's a pity that the person who downvoted my answer didn't explain the reason for that. $\endgroup$ – José Carlos Santos Sep 17 '17 at 10:27
  • $\begingroup$ +1: my rep on this part of stack won't show my vote. I'm torn between your answer and Mauro's. $\endgroup$ – Lugh Sep 18 '17 at 2:49
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    $\begingroup$ This, in my opinion, is one of many possible answers to a different question. $\endgroup$ – Francois Ziegler Sep 20 '17 at 4:58
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There are some borderline cases. Frege wrote a two volume treatise on set theory, Die Grundlagen der Arithmetik [The Foundations of Arithmetic] (1884). Russell found a contradiction in Frege's system (Russell's paradox). So mathematicians came up with other systems for doing set theory.

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  • $\begingroup$ One should briefly point out in this context that Frege-style proof systems are studied to this day with a view towards their complexity. One should also make it clear that: (0) Bertrand Russell's name has two l's, and (1) the inconsistency of the axiom scheme of unrestricted comprehension is quite separate and irrelevant to Frege's propositional calculus I am not writing this to dispute that Russell found an error; (2) strictly speaking, OP's title asked for errors in proofs. $\endgroup$ – Peter Heinig Sep 17 '17 at 8:17
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    $\begingroup$ On second though, I can't see how the example of Frege and Russell is relevant to what the OP is asking for. Russell pointed out a logical mistake, modulo classical logic. The OP seems to be asking for failed applications of abstract theories. A bridge built with too much blind trust to some abstract method of calculating certain parameters, and soon after having collapsed, would be an example, yet I don't know of such an example. $\endgroup$ – Peter Heinig Sep 17 '17 at 10:11
  • $\begingroup$ @PeterHeining I am familiar with Russel's paradox, and had a similar reaction; however, of the two questions I had recieved within roughly 48 hours, and how the other left me discouraged, I... acted hastily in accepting it. $\endgroup$ – Lugh Sep 18 '17 at 2:43
  • $\begingroup$ @Lugh: sorry for the pedantry, but it's "Russell". Two 'l's, that is. $\endgroup$ – Peter Heinig Sep 18 '17 at 6:00
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Proofs (your title) in mathematics (your tag) don’t get invalidated. Some of Kant’s more exalted efforts, on the other hand... (See e.g. Prop. 4 or 7 in his Metaphysical foundations of dynamics.)

A famous quote comes to mind [german, english, context] (emphasis mine):

At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?

In my opinion the answer to this question is, briefly, this: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. It seems to me that complete clearness as to this state of things first became common property through that new departure in mathematics which is known by the name of mathematical logic or “Axiomatics.” The progress achieved by axiomatics consists in its having neatly separated the logical-formal from its objective or intuitive content; according to axiomatics the logical-formal alone forms the subject-matter of mathematics, which is not concerned with the intuitive or other content (...)

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  • $\begingroup$ Proofs in mathematics get invalidated if mathematics is the whole business, not only the "not yet refuted theory". See my answer concerning the Schröder-Bernstein Theorem. $\endgroup$ – Wilhelm Aug 29 '18 at 14:06
  • $\begingroup$ In @Wilhelm’s answer, people saying “this is no proof” recognize it never was. This is as it should be, and totally different from “observed phenomena invalidating an a priori proof” because “mathematics requires a posteriori verification.” Of that, no example has yet been given here. $\endgroup$ – Francois Ziegler Sep 1 '18 at 21:09
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Read this link for a relatively recent example:

In Mathematics, Mistakes Aren’t What They Used To Be
by Siobhan Roberts

[Voevodsky] and Kapranov shared a passion for developing the mathematics of new higher dimensional objects and categories, and they published an important result in 1990. [...]

But along the way he met a bump in the road. In 1998, the American mathematician Carlos Simpson published a paper indicating there might be a mistake in Voevodsky and Kapranov’s 1990 result. For years Voevodsky sifted through the details without making much progress. He remained convinced the result was right. Then, in the autumn of 2013, as the leaves changed color and summer gave way to autumn, he made a breakthrough. Of sorts. He confirmed the error. The important result was no longer quite so important.

“It is plainly wrong. The main theorem is incorrect,” he says. “It’s not that there is some gap in the proof. It’s that the main theorem is plainly wrong.” The mistake, he explains, was in failing to question the obvious. “We had proved that an assertion was indeed true in all of the difficult cases, but it turned out to be false in the simple case. We never bothered to check.” In confirming the error, he added an addendum to the original citation in his official publications list—“Warning: The main theorem of this paper was shown by Carlos Simpson to be false.”

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The Schröder-Bernstein Theorem has a remarkable history. Repeatedly stated (and claimed as proven) between 1882 [G. Cantor, letter to R. Dedekind (5 Nov 1882)] and 1895 [G. Cantor, collected works, p. 285] but never really proved by Cantor, this theorem is called after Ernst Schröder and Felix Bernstein, because both proved it. Alwin Korselt however discovered a flaw in Schröder's proof in 1902. Alas the Mathematische Annalen did not publish the correction before 1911. [A. Korselt: "Über einen Beweis des Äquivalenzsatzes", Math. Ann. 70 (1911) 294] Nevertheless it took some time until this correction received public attention. Ernst Zermelo noted in his edition of Cantor's collected works as late as in 1932: "The theorem, here mentioned without proof [...] has been proved only in 1896 by E. Schröder and 1897 by F. Bernstein. Since then this 'equivalence-theorem' is considered of the highest importance in set theory." [G. Cantor, collected works, p. 209] We learn from this that wrong proofs can survive in mathematics over many decades.

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In this paper (in French) Mémoire sur les systèmes d'équations aux dérivées partielles d'ordre quelconque et sur leur réduction à des systèmes d'équations linéaires du premier ordre Cauchy claimed that every analytic semi-linear Partial differential equation with analytic data has a unique solution, which is shown here to be false by Sofia Kovalevskaya

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