Questions tagged [mathematical-logic]

For question regarding the applications of formal logic to mathematics.

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Claims that fully formal proofs are impossible to write down

It is sometimes asserted that human-readable mathematical proofs that we construct and publish are just informal approximations to the gold standard, which is a completely formal proof in a formal ...
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Formal logic as a synonym to syllogistic logic, or as a name for the study of logic?

On page 443, section 1.1 Expanding to Contradiction, in José Ferreirós' A Road To Modern Logic - An Interpretation, the following is written: Philosophical conceptions of logic have been complex and ...
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The definition of 'countable' and 'finite' set

I have an equivalent definition of finite and countable sets that seems 'less' based on set theory than the usual definitions. I am wondering if anyone has encountered these definitions anywhere, esp....
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Context of the discovery of ultraproducts

Łoś's theorem is a fundamental theorem in model theory (a branch of mathematical logic). Historical question: What was Łoś's original motivation to define ultraproducts and prove Łoś's theorem? Which ...
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Frege: Defining a universe of ordered pairs?

Does anyone know how to define the set (or class) of all ordered tuples? In Frege's Begriffsschrift a tuple $(x, ..., z)$ is defined as a primitive object which is provided as an argument to a ...
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1 answer
156 views

Relation between Bourbaki group and Vienna Circle

Background: I'm an undergraduate student in Mathematics and I study Mathematical Logic and Philosophy of Science in an undergraduate research project (here in Brazil, where I'm from, we call this kind ...
2 votes
2 answers
139 views

Earliest proof of the soundness of first order predicate logic

When was the soundness theorem for first order predicate logic (quantification theory) first proven? Is there any evidence that soundness was presupposed or taken as self-evident prior to 1930?
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1 answer
146 views

Is multiplication postulated axiomatically in Peano arithmetic?

I figured this question is better suited to this stackexchange. I give some mathematical details, but this is primarily an HSM question. According to this post, the existence of multiplication in ...
3 votes
1 answer
199 views

Checking a Gödel 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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Do units in physics have any link to the origins of type theory in logic?

It is well known that the theory of types, first introduced by Bertrand Russell in 1903 and developed with Whitehead in their Principia Mathematica (1910), was a way to deal with paradoxes in set ...
4 votes
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269 views

After the invention of Boolean algebra, why did it take so long for the existential and universal quantifiers to become formalized into symbols?

Was there a specific reason that prevented researchers in Boolean algebra to invent such quantifiers in the flexible format that are known today earlier? Since the compact symbols for multiplication ...
1 vote
1 answer
207 views

The status of Tarski's theory of truth

Tarski's truth theorem asserts that a truth definition for a (reasonably strong) theory cannot be formalized within that theory. I have seen that Tarski's theory of truth has received a lot of ...
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211 views

What are some notable contributions of mathematical logic to mathematics (outside of mathematical logic)?

I have been reading an introductory text in mathematical logic (Holden, 1995). The final chapter presents the resolution of Hilberts's tenth problem concerning the integer roots of an arbitrary ...
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How much did scholasticism contribute to logic and mathematics?

I occasionally encounter mentions suggesting that late middle ages weren't as barren intellectually as commonly believed. For example, Occam and Scotus are credited with developing modal and ...
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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 ...
3 votes
3 answers
283 views

History of Foundation of Mathematics

I know that studying the evolution and history of a certain subject is a way of resolving the complexity of that subject. So I want to ask about references that describe the history and evolution of ...
6 votes
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578 views

A branch of mathematics which refused to be rigorous?

I'm currently in a class on formal mathematics/formal logic/axiomatic set theory. Someone asked, "At the end of the day, as mathematicians, why do we care about rigor?" My professor gave an ...
13 votes
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595 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 ...
2 votes
2 answers
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When was compactness theorem for propositional logic first proven?

Compactness for first-order predicate logic was first proven as a corollary of (Gödel 1930). Does anyone know a reference for the first proof of the compactness of propositional logic? Some proofs ...
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148 views

Origins of Stone duality

My question is a mix of mathematical and historical, if you consider my question will be better answered in the mathematics community, please tell me. I want to know the historical roots of Stone's ...
2 votes
1 answer
115 views

Why was there a thirty year gap between the compactness theorem and nonstandard analysis?

Why was there an approximately thirty year gap between the discovery of the compactness theorem (for countable theories) in 1930 and Robinson's elucidation of nonstandard analysis in the early 1960s? ...
5 votes
1 answer
290 views

How did the principle of explosion come up and was developed historically? [closed]

I am looking for some background information on the principle of explosion. When was it first formulated, how was it justified, and how did it affect interpretation and development of logic? Please ...
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1 answer
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What are the Peirce's axioms of arithmetic and how do they relate to the Peano axioms?

I will be glad if someone who has seen Peirce's paper could summarily describe here Peirce's axioms and describe their relation to Peano's.
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Who first estimated the minimal size of a model for a satisfiable formula of monadic first-order logic?

Lowenheim (1915) is credited with the first proof of the decidability of semantic validity over the monadic fragment of first-order predicate logic. However, I find no reference in his proof to an ...
2 votes
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Kurt Gödel: a biography

I'm looking for a well-written biographical book about Kurt Gödel. Any titles you'd recommend?
4 votes
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Why are there so many different, and widely accepted, notational systems for boolean logic?

I can write out the following CNF in various different ways: In mathematical textbook notation: $(A \land B \land C) \lor (\lnot A \land B \land \lnot C) $ In C-like programming notation: ...
2 votes
1 answer
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Was C.S Peirce in his abductive reasoning influenced by the generalization from special cases that is used by mathematicians?

I am reading some random bits from Peirce's collected works and they give me the impression that Peirce tried to integrate every(or nearly every) major scientific and mathematical concept which he was ...
4 votes
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What is the basic idea behind calculation of area? [closed]

The system of calculating area in terms of square units is pretty philosophical and not very intuitive. It must have taken a great amount of time for humanity to arrive at such a convention and to ...
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What were C.S Peirce's contributions to theory of relations?

In my study of Peirce I came across a number statements that he extended De Morgan's relation algebra. What exactly are his contributions to relation algebra?
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107 views

Where is First-Order Peano Arithmetic first clearly formulated?

I really should know this, but ... When/where/by whom was first-order Peano Arithmetic first clearly and explicitly formulated in a recognizably modern form (perhaps exact notation apart) -- with the ...
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3 answers
88 views

Where does the name "geometric theory" come from?

In mathematical logic, where does the adjective "geometric" comes from, in terms like "geometric theories" and "geometric logic"? These terms come up in fields like topos ...
13 votes
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558 views

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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Who defined $A\to B$ that yields false whenever $A$ is true and $B$ is false and why was it defined that way? [duplicate]

In logic we are taught the following definition. $$A\to B$$ that yields false whenever $A$ is true and $B$ is false. Who defined this and why was it defined that way? What inspired him/her?
4 votes
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608 views

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 ...
7 votes
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431 views

Where does the material implication come from, if not from George Boole?

I looked through both of George Boole's treatises (1 and 2), but there is nothing like implication as I have seen it, with $$F \rightarrow F = T$$ $$F \rightarrow T = T$$ $$T \rightarrow F = F$$ $$T \...
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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 ...
3 votes
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Translations of "Sur le théorème de Zorn"?

Are there any translations of the following into English, German, or Russian? Nicolas Bourbaki, Sur le théorème de Zorn, Archiv der Mathematik, Volume 2, pages 434–437, November 1949. Any help is ...
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Hilbert's reaction to Gödel's incompleteness theorems

Is it known how Hilbert initially reacted to Gödel's incompleteness theorems upon their announcement at the Königsberg conference in 1930, or their publication in 1931?
10 votes
4 answers
787 views

How much of mathematics did Russell's paradox really break?

According to my quite likely wildly oversimplified understanding, a revolution occurred in the foundations of mathematics when Cantor's formulation of set theory was found to be inconsistent due to ...
3 votes
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How did Kolmogorov came up with his formalization of intuitionistic logic?

According to this article Kolmogorov published a paper in 1925 in which he attempted to formalize Brouwer’s intuitionistic mathematics. In that paper there are the following logical formulas: \begin{...
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In which article/book chapter did Cantor, Hibert, and Poincare formally defined or directly discussed the term “potential infinity”?

Some media sources say that "Cantor claimed that there would only be potential infinity, not actual infinity" In addition, the following link claims that Hilbert, Poincare, and Cantor were ...
7 votes
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Why is the existential quantifier symbol ∃ a backwards "E"?

Peano introduced a number of logical symbols still used today: $∨$ (from Latin vel) $∧$ (inverted $∨$) $∃$ This inversion of Latin letters as symbols (and inversion of symbols to signify their '...
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Who superseded Peano's dot notation in symbolic logic and when?

Bertrand Russel gave an exhaustive treatment of creating mathematics from logic in Principia Mathematica (1910-1913), using the logical notation created by Frege and Peano. As monumental as this is, I ...
6 votes
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Was Von Neumann and Birkhoff's original formulation of Quantum Logic related with projective geometry?

I was looking at how did von Neumann and Birkhoff formulate their Quantum Logic formalism back in 1936. To solve some questions, I contacted via email a philosopher who studied this topic. I thought ...
14 votes
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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]...
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What did Hans Bethe think of von Neumann's quantum logic?

Nobel laureate Hans Bethe was a friend of mathematician-physicist John von Neumann, and he once said: "I have sometimes wondered whether a brain like von Neumann's does not indicate a species ...
3 votes
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Where does the notion of "three crises of mathematics" come from? [duplicate]

Update: It can be traced back to Fraenkel-Bar-Hillel's Foundations of Set Theory, originally published in 1958. Further discussions can be seen at the linked question. The notion of "three crises ...
2 votes
2 answers
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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. The related question most people ask is: why does $-\times-=+$. Of, course there may be several explanations for ...
1 vote
1 answer
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Why is Robinson arithmetic "Q"?

I see Peano arithmetic so often abbreviated as "P" or "PA". Why is Robinson Arithmetic "Q"? Following the obvious pattern, I would have expected R" or "RA".
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When did people start to state and justify properties of arithmetical operations?

I have question regarding the history of the idea of founding mathematics (specially arithmetic) on a logical basis. What I'm interested in knowing is, at what point historically people started to ...