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20 votes
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What were 12 year old Pitts' objections to Principia Mathematica?

This story bears characteristic signs of a tall tale, although in this case one can identify the origin. It appears to be an amalgamation of two anecdotes, neither of which is itself very credible. ...
Conifold's user avatar
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17 votes

How did Russell arrive at the paradox demonstrating the inconsistency of naive set theory?

Russell was not the first to discover "his" paradox. By June 1901 when he arrived at it (it was not published until the first edition of Principia in 1903), it was already known for a while ...
Conifold's user avatar
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16 votes

Why did Hilbert believe consistency implies existence?

Hilbert wrote this observation some decades before the formal theories of completeness (and incompleteness) took shape, so you can't expect him to be overly precise with regard to such notions. The ...
Mikhail Katz's user avatar
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13 votes
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Was Paul Cohen a student or assistant of Gödel?

No, he was not. Cohen wrote his own account of the history of forcing, The Discovery of Forcing (Rocky Mountain J. Math. 32 (4) (2002), 1071-1100), where he addresses his relationship with Gödel in ...
Conifold's user avatar
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11 votes
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What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

Vitali's construction is probably the first example "unpleasant" in the modern sense, but well-ordering of the continuum was unpleasant enough to some at the time of Zermelo's proof in 1904. Accepting ...
Conifold's user avatar
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10 votes

Who introduced the notation $y|_{x=a}$?

You can see : Giuseppe Peano , Lezioni di Analisi Infinitesimale, 2 vols., 1893, page 17 : $$[f(x)]_{x=a}=f(a).$$ Not sure it is the earliest... but Peano was a prolific "inventor of notations". ...
Mauro ALLEGRANZA's user avatar
10 votes

How could the people of the past be sure that $a \times b = b \times a$?

Multiplication, before the invention of modern (axiomatic) algebra, was defined as the operation giving the area of a rectangle with sides of a particular length.1 Commutativity of multiplication ...
Jonathan Cast's user avatar
10 votes
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Who coined the term "iff" for "if and only if"?

As Francois Ziegler notes, Kelley attributes it to Halmos. In the past (before Halmos), definitions might be given in the form A group is called abelian if $xy=yx$ for all $x,y$. ...and every ...
Gerald Edgar's user avatar
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10 votes
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Where does the material implication come from, if not from George Boole?

It may be surprising, but the material implication does not come from truth tables, the truth table definition is a late development. Neither de Morgan, nor Peirce, nor Frege, nor even Russell came up ...
Conifold's user avatar
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10 votes
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A branch of mathematics which refused to be rigorous?

You might be looking for the Italian School of Algebraic Geometry. It has become the canonical example of problems with a lack of rigour. The short summary is that the school started with some ...
cesaruliana's user avatar
9 votes
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Really confused about the history of logic / proofs / etc

Mathematical logic is a quite modern discipline : it emerged in the mid-19th century with Boole, Peirce and Frege. Logic instead, is quite ancient : we can date it at least from Aristotle (384–322 ...
Mauro ALLEGRANZA's user avatar
9 votes
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Why is the existential quantifier symbol ∃ a backwards "E"?

When introducing the older terminology in the previous sentence, Peano describes it thus: ... signifie "il y a des a", "les a existent"... It seems likely this is the source of the inverted "E".
iacob's user avatar
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8 votes
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When did mathematicians first use the contrapositive form to prove a conditional statement?

This is one of those cases where it is easy to take modern ideas for granted, and wonder how something so "obvious" could be missed. Modern formal logic with its array of logical operations, ...
Conifold's user avatar
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8 votes

Checking a Gödel quote on Principia Mathematica

In responding to your question, I hope that I can convince you of the following two claims: Gödel held that the formal system of the actual Principia was that of ramified type thoery (simple type ...
Landon D. C. Elkind's user avatar
8 votes

What brought about the need for real analysis and formal logic in recent years?

As I explained in my answer to your other question, mathematics was always done using ordinary (non-formalized) logic. Attempts to formalize logic begin with Aristotle. (This is called "formal logic")....
Alexandre Eremenko's user avatar
8 votes

Has the standard of mathematical proofs changed over time?

Yes and no. It is better to say that there were always several different standards. Most proofs of Euclides, Archimedes and Apollonius are on the level of modern standards, though gaps in those proofs ...
Alexandre Eremenko's user avatar
8 votes
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Who superseded Peano's dot notation in symbolic logic and when?

This is not so straightforward, see Peirce, Frege, the Logic of Relations, and Church's Theorem by Dipert for a sketch of history. The notation Russell used was not created by Peano, and certainly not ...
Conifold's user avatar
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8 votes

Why did Hilbert believe consistency implies existence?

I just want to add some context to Katz's very nice answer. Hilbert's work on foundations occurred in the aftermath of the intuitionistic criticisms by Brouwer and Weyl. (Weyl in particular must have ...
Michael Weiss's user avatar
7 votes
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Did Gödel know about Turing degrees in 1946?

I do not believe that degrees of computability were directly relevant to the topic of Gödel's 1946 lecture because they do not relativize computability in the relevant language dependent sense that ...
Conifold's user avatar
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7 votes
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Could the incompleteness theorems have been proved by Tarski if Gödel had not proved them first?

In principle, yes. See: Roman Murawski, Undefinability of Truth. The Problem of Priority: Tarski vs Gödel (1998): Tarski made clear his indebtedness to Gödel's methods. [...] On the other hand ...
Mauro ALLEGRANZA's user avatar
7 votes

Who was the first to write proofs in this manner?

Whitehead & Russel, Principia Mathematica (1910) is pretty much like this. The justifications are on the left, and only certain lines are numbered. But I doubt it is the first. Compare Frege's ...
Gerald Edgar's user avatar
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6 votes

Who introduced the notation $y|_{x=a}$?

Concerning alternative notations for $y|_{x=0}$: Lagrange in Théorie des fonctions analytiques, 1797, p.57 writes: (...) et si on désigne par $(y), (y'), (y''),$ etc. les valeurs de $y,y',y'',$ ...
Michael Bächtold's user avatar
6 votes

History of Foundation of Mathematics

Some references : Morris Kline, Mathematics: The Loss of Certainty (1982) Marcus Giaquinto, The Search for Certainty : A Philosophical Account of Foundations of Mathematics (2002) José Ferreiròs, ...
Mauro ALLEGRANZA's user avatar
6 votes

Did Gödel consider himself primarily a philosopher who is interested in mathematics (instead of the other way round)?

Kurt Gödel enrolled in the University of Vienna, attending lectures on physics, his initial field of interest, lectures on philosophy given by Heinrich Gomperz, and lectures on mathematics. [...] ...
Mauro ALLEGRANZA's user avatar
6 votes

What brought about the need for real analysis and formal logic in recent years?

We didn't seem to have a "proof theory" where we all agreed what constituted a proof or what was considered a correct / incorrect proof. Yes we did. (The Greeks theorized proof by contradiction, ...
Francois Ziegler's user avatar
6 votes
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When was compactness theorem for propositional logic first proven?

For a general overview of the history of first order logic see SEP, The Emergence of First-Order Logic. On the history of compactness theorem more specifically see Dawson, The compactness of first-...
Conifold's user avatar
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6 votes

In which article/book chapter did Cantor, Hilbert, and Poincaré formally define or directly discusse the term “potential infinity”?

You have misinterpreted the article you refer to; nowhere does it say that "Cantor claimed that there would only be potential infinity, not actual infinity". In fact, it says the opposite: ...
Spencer's user avatar
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6 votes

Earliest proof of the soundness of first order predicate logic

See D.Hilbert & W.Ackermann, Principles of Mathematical Logic: the 1950 American translation of the 1938 second edition of Grundzüge der theoretischen Logik. The 1928 first edition is considered ...
Mauro ALLEGRANZA's user avatar
6 votes
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Relation between Bourbaki group and Vienna Circle

Short answer: not much of a relation. Perhaps the stars could have aligned differently if the Vienna Circle lasted longer, but with the rise of Nazism many members had to flee Austria and the meetings ...
Conifold's user avatar
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6 votes
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Has Penrose ever acknowledged criticism of the Penrose-Lucas argument?

Penrose responded to various commentaries made regarding his 1994 book "Shadows of the Mind" in his Beyond the Doubting of a Shadow. I never spent much time reading over this since my ...
Mark Yasuda's user avatar
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