20
votes
Accepted
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. ...
- 80k
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 ...
- 80k
17
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 ...
- 6,683
13
votes
Accepted
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 ...
- 80k
12
votes
Accepted
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 ...
- 80k
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".
...
- 15.2k
10
votes
Accepted
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 ...
- 10.6k
10
votes
Accepted
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 ...
- 80k
10
votes
Accepted
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 ...
- 898
9
votes
Accepted
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 ...
- 15.2k
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 ...
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")....
- 51.1k
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 ...
- 51.1k
8
votes
Accepted
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".
- 370
8
votes
Accepted
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 ...
- 80k
8
votes
Accepted
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 ...
- 80k
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 ...
- 4,922
7
votes
Accepted
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 ...
- 80k
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 ...
- 10.6k
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
$y,y',y'',$ ...
- 1,824
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, ...
- 15.2k
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, ...
- 7,809
6
votes
How did the principle of explosion come up and was developed historically?
The principle/law of explosion had a curious history. Anellis in his review of Handbook of the History of Logic suggests that it was known already to ancient Stoics. Indeed, it is closely linked to ...
- 80k
6
votes
Accepted
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-...
- 80k
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:
...
- 602
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 ...
- 15.2k
6
votes
Accepted
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 ...
- 1,558
6
votes
When did mathematicians realize that theory of algebraically closed fields admits quantifier elimination?
A useful sketch of history is given in Alfred Tarski's Elimination Theory for Real Closed Fields by van den Dries. The result was known to Tarski by 1948 (when his Decision Method for Elementary ...
- 80k
5
votes
Accepted
A 'third way' to consistency (besides Dedekind's and Gentzen's)? Hilbert's 'unverstanden' 1904 Heidelberg ICM proposal for consistency proofs
In regards to your bibliographic question, I suggest the highly informative essay by Craig Smoryński, Hilbert's programme; he gives a highly compressed account of Hilbert's 1904 proof on pp.7-8 of the ...
- 383
5
votes
Where does the material implication come from, if not from George Boole?
Charles Sanders Peirce is credited with the introduction of truth tables in an unpublished manuscript dated 1893. This includes a truth table for what we now call material implication. A detailed ...
- 7,237
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