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Questions tagged [mathematical-logic]

For question regarding the applications of formal logic to mathematics.

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Has the standard of mathematical proofs changed over time?

Why I asked this question : https://gallica.bnf.fr/ark:/12148/bpt6k90195m/f54.image p 50-51, in course of Cauchy, a proof of the intermediate value theorem. Now, that's not a proof. And I learned ...
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What brought about the need for real analysis and formal logic in recent years?

I can't seem to find a clear, definitive, non-circular answer on this. For centuries and centuries, we've been doing mathematics in one form or another, be it geometry and pictures, or inventing ...
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133 views

Really confused about the history of logic / proofs / etc

I'm a little confused when I read that mathematical logic is actually a very recent field (1800's - 1900's), regarding the foundations of mathematics and so on. By this I refer to writing proofs in ...
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1answer
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What does Rousseau mean by “Baroco des Logiciens”?

In the Wikipedia "Baroque" article I found this quote from "Dictionnaire de Musique" by Jean-Jacques Rousseau: BAROQUE. Une Musique Baroque est celle dont l’Harmonie est confuse, chargée de ...
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Question about Leibniz's “characteristic numbers” and propositional logic

The wikipedia article on Gottfried Wilhelm Leibniz mentions, in the chapter on symbolic thought, that: "Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers ...
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150 views

Checking a Godel quote on *Principia Mathematica*

Is there serious doubt of whether the first edition of Russell and Whitehead's Principia Mathematica used the ramified theory of types? I am travelling and cannot easily check sources but I do easily ...
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180 views

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

In Wikipedia on the page dedicated to Walter Pitts (accesses today), it is written that, He is widely remembered to have spent three days in a library, at the age of 12, reading Principia ...
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1answer
106 views

Was Paul Cohen a student or assistant of Gödel?

In The Man Who Loved Only Numbers, a biography about Paul Erdős, by Paul Hoffman, the author claims that Paul Cohen was "Gödel's former assistant" (p 225). However, I can't find any other sources ...
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How did Wittgenstein fulfill eligibility requirements for a PhD in philosophy without having a Bachelor's degree in philosophy?

The Wikipedia article about Wittgenstein says: In Norway it was clear that Moore was expected to act as Wittgenstein's secretary, taking down his notes, with Wittgenstein falling into a rage when ...
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Who was the first to use the “does not exist” sign ∄?

Who was the first to use the "does not exist" sign ∄? I'm aware that Giuseppe Peano originated serifed ∃ and, moreover that Whitehead and Russell repurposed Peano's serifed ∃; I'm also aware that ...
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The Principia Mathematica's missing chapters

The numbering of the chapters of Bertrand Russell and Alfred Whitehead's Principia Mathematica is the following : 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 30, etc overall, ...
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How were variables used and understood in (particularly) 19th century maths?

Context: I have been thinking about Frege's Begriffsschrift, where he introduces a version of what we now think of as the standard quantifier/variable notation. Philosophers who write on Frege tend to ...
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337 views

Who coined the term “iff” for “if and only if”?

The OED's entry for "iff" lists this as the earliest usage: 1955 J. L. Kelley Gen. Topol. vii. 232: "F is equicontinuous at x iff there is a neighborhood of x whose image under every member of ...
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Is there a formal distinction between potential and actual infinities?

In modern set theory the difference between actual infinity and potential infinity is often not understood or even denied. Some decades back however mathematicians like Hilbert or Poincaré, let alone ...
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1answer
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Did Euclid formulate his definitions/postulates/common before or after writing all his theorems?

Did Euclid formulate his definitions/postulates/common notions at the beginning of Book I of the Elements before or after writing the 465 theorems of the Elements? cf.: Michael J. Crowe, “Ten ...
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1answer
161 views

Did Gödel know about Turing degrees in 1946?

A much quoted passage from Gödel is the opening section of his Remarks before the Princeton bicentennial conference on problems in mathematics (1946) where he praises Turing's Turing machine model of ...
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Gentzen and computer science

This is a cross-post from mathstack: https://math.stackexchange.com/questions/2584003/gentzen-and-computer-science?noredirect=1#comment5333947_2584003 I would like to learn a bit about the ...
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1answer
108 views

Why second-order logic?

Wikipedia says: First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For ...
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How did Aristotle influence Euclid?

In other words, how is Aristotle's logic represented in Euclid's Elements? I have read many articles where Euclid's Elements is linked to Aristotle's logic, but I do not understand, and I can't find ...
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Timeline of mathematical foundation?

As it is globally known that set theory as a foundation of mathematics, although in the beginning we didn't call it "Set" rather group of elements. For example - set of [1(banana) + 2(apple)+1(cow)] =>...
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What are the paradoxes of which Russell is talking about in the following?

I was reading Russell's An Inquiry into Meaning and Truth (1940) and I noted the following passage in the Introduction, Finally, there is the question of the relation between truth and knowledge. ...
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A 'third way' to consistency (besides Dedekind's and Gentzen's)? Hilbert's 'unverstanden' 1904 Heidelberg ICM proposal for consistency proofs

Question. (bibliographic) Are there recommendable, preferably modern, detailed and technical discussions in the literature on the kind of consistency proofs that Otto Blumenthal mentions Hilbert ...
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What are early examples of the rare notational convention to make the sign of the real number represented by a letter depend on the typography?

Question. What early published or citably attested examples (preferably in the mathematical literature) can you give of the following convention? Let $\mathbb{S}$ denote some nonempty subset of some ...
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1answer
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How did people realize they could do logic with electronics?

How did people realize they could do logic with electronics? Are there anecdotes of the first realizations? I'm wondering about the first "eureka" moments.
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173 views

Why and who was the first to denote the square root operation in fractional form as $1/2$

Basically, the square root operation was discovered and proved rigorously from the Pythagorean theorem, it was denoted by square root of a rational number say $n$ as $\sqrt{n}$, but at a later stage, ...
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Has the modern logic negation $\lnot$ been adapted from Frege's Begriffsschrift?

Has the modern logic negation $\lnot$ been adapted from Frege's Begriffsschrift?
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114 views

Books on recent developments of abstract mathematics

I've read Marcus Du Sautoy's "The Music of Primes" recently and, although I'm not so much interested in number theory, I really liked it for the global perspective on the subject, though at an "...
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Carnap's last theory Of probability

According to Bar-Hillel, Carnap's coauthor in a 1952 report on probability, Carnap had, as of 1956 an unpublished but circulated theory distinguishing "random" refers to methods of production of ...
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1answer
133 views

Measurable cardinals and the fundamental theorem

Who was the first mathematician to define and investigate measurable cardinals? And how long did it take for the fundamental theorem of measurable cardinals to be hypothesized and proven? I would be ...
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3answers
165 views

History of Foundation of Mathematics

I know that studying the evolution and history of a certain subject is a way of resolving Complexity of that subject .. so i want to ask about references that describe the history and evolution of ...
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What is the relationship between constructivism and object oriented programming?

I am exploring the topic of constructivism from discreet math, and think it is related to object oriented programming. Can anyone confirm or deny the two are related?
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Who introduced the notation $y|_{x=a}$?

When a variable $y$ depends on other variables, say $y=c x^3$, one often writes $$y|_{x=2}$$ to say "$y$ when $x$ has value $2$". This might be more familiar in the context of derivatives where we ...
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What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

I read that Zermelo formulated AC in 1904 in order to formally prove the well-ordering theorem. Vitali’s 1905 proof of the existence of a non-measurable set of real numbers appears to the first “...
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2answers
269 views

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

As described here, we know that: In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted ...
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Did Gödel consider himself primarily a philosopher who is interested in mathematics (instead of the other way round)?

Kurt Gödel is one of the "best" logicians of the 20th century. Here, the user "Jeffrey Kegler" states that: Kurt Gödel considered himself a philosopher who did mathematics, rather than as a ...
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How could the people of the past be sure that a * b = b * a?

Let me quote from Terence Tao "Analysis 1": Histocially, the realization that numbers could be treated axiomatically is very recent, not much more than a hundred years old. Then, how could the ...
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2answers
255 views

When did mathematicians first use the contrapositive form to prove a conditional statement?

I am reading parts of Euclid’s Elements and I am surprised, rightly or wrongly, to see that Euclid did not recognize that a conditional is logically equivalent to its contrapositive form. Indeed, one ...
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What was Gödel's proof that errors in U.S. constitution could allow the U.S. to become a fascist dictatorship?

According to this document, Kurt Gödel had discovered logical inconsistiencies in the U.S. constitution that could theoretically allow it to become a dictatorship. Morgenstern told him not to bring it ...
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Could the incompleteness theorems have been proved by Tarski if Gödel had not proved them first?

As I know, Tarski had proved his undefinability theorem by $1936$, $5$ years after Gödel's incompleteness theorems had been discovered. I wonder whether his original proof was built upon the work of ...
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1answer
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Was the idea of encoding meta-mathematics into arithmetic in the air when Gödel proved incompleteness theorems?

I mean, was there anyone who tried (before Gödel) to encode meta-mathematics? Or, was the idea of constructing formal sentences which informally refer to itself completely new when Gödel introduced ...
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When or why & who originated this puzzle, $0.999… = 1$ [duplicate]

The problem is the infinite or endless repeated digits of $9's$ after zero digit and the decimal notation, Despite its apparent simplicity & the huge talk about it every where in mathematics or ...
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Who said $\pi$ is a constant since it is not even a real number?

EDIT: (130116) I don't mean it is complex or imaginary nor it is negative also, I tried hard to conceive it on the real line number (positive X-axis), by obvious means, a little idea came to me?, "...
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2answers
393 views

Who wrote down minus times minus is equal to plus? [duplicate]

I am not here to ask why minus times minus is plus, this is a basic arithmetic fact. Most people asks that why $-\times-=+$ and ofcourse there may be several explanations to this fact. But I want to ...
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1answer
107 views

History of Mac Lane's dissertation on abbreviated proving

Saunders Mac Lane published his thesis (Abgekuerzte Beweise in Logikkalkul) which anticipated to some degree constructing proofs of theorems by constructing programs, giving a sufficient input, and ...
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1answer
135 views

Timeline of Leibniz' Propositional Logic

Gottfried Leibniz developed a system of propositional logic in the late 1600's, which wasn't published until 1903, when it was discovered in the Royal Library of Hanover by Louis Couturat. How did ...
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Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?

Kontinuum is German for continuum, but Cantor used $\mathfrak{c}$. Revision. J.W.Perry questions whether or not Cantor ever in fact used the symbol $\mathfrak{c}$. I must admit I just assumed that he ...
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Mathematical induction only dates from the Middle Ages? [duplicate]

The technique of "mathematical induction" is a method of proof where you show some theorem is true for some starting integer and prove also that it holding at any arbitrary integer implies it must ...
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Why is ZFC used more widely than NBG?

In studying the solutions proposed for Russell's Paradox on naive set theory (mainly the corresponding entry in the Stanford Encyclopedia of Philosophy) I came across the following considerations: von ...
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1answer
250 views

Why does logical-AND take operator precedence (evaluated first) over logical-OR?

Does logical-AND have precedence over logical-OR because of a reason or was it an arbitrary choice made sometime in the distant past? (Perhaps it could have been the other way around: OR-terms ...
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3 Poles and 3 Texans who had read “Principia Mathematica”

To quote Bertrand Russell, "My Philosophical Development", Simon and Schuster, N.Y., 1959, p. 86: I used to know of only six people who had read the later parts of the book [Principia Mathematica]...