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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 ...
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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 ...
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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 ...
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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 ...
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Claims that fully formal proofs are impossible to write down

In Donald Mackenzie's article, The Automation of Proof: A Historical and Sociological Exploration, I found some statements by P. Nidditch that come close to what I have been looking for. In the whole ...
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How could the people of the past be sure that $a \times b = b \times a$?

The guesses that are ventured in this question are entirely wrong. (1) That our reason to think multiplication is commutative is that we have an axiom that says so is wrong. (2) That those without ...

Why did Hilbert believe consistency implies existence?

Think back to when you learned linear algebra: \begin{align} x + y + z & = 2 \\ 6x - 4y + 5z & = 31 \\ 5x + 2y + 2z & = 13 \end{align} Does a solution to this system exist? (In this case, ...
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Who first proved that empty set is subset of all sets?

Giuseppe Peano, Studi di Logica matematica (1897): Il segno $\Lambda$, fra classi, indica la classe nulla, cioè non contenente alcun individuo. Si può definire come segue [The $\Lambda$ symbols ...
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