Questions tagged [mathematical-logic]
For question regarding the applications of formal logic to mathematics.
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Did Grothendieck have any thoughts on foundations of mathematics? [closed]
I remember reading that Grothendieck didn't care much about foundational issues and didn't want to be 'stuck all the way down there'.
Does anybody know if he ever actually said this? Did he have any ...
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When were Peano axioms formulated purely in first-order logic?
It is mentioned on the Wikipedia article about Peano axioms that:
The ninth, final axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes ...
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Universal logicians
Henri Poincaré is often considered to be the last universal mathematician, meaning the last individual who made contributions across all areas of mathematics of his time. Today, even a prodigious ...
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Role of Alessandro Padoa in the development of modern mathematics
Here is an excerpt from ESSAI D’UNE THÉORIE ALGÉBRIQUE DES NOMBRES ENTIERS, PRÉCÉDÉ D’UNE INTRODUCTION LOGIQUE A UNE THÉORIE DÉDUCTIVE QUELCONQUE from Alessandro Padoa (as can be found here):
nous ...
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Why isn't Boethius's Thesis more commonly accepted in mathematics and logic?
Why isn't Boethius's Thesis, that the negation of an implication is another implication where the consequent is negated, a commonly accepted axiom in mathematics and logic? It is an axiom of connexive ...
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Medieval Logical Interpretations of the Word "All"
Paraphrasing the Philosopher (Aristotle): Forms of speech are either simple or composite. If expressions are simple, then they are neither true nor false. With this said, did some Medieval logicians ...
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Who first introduced semantic and syntactic consequence
The relations $A \vdash B$, read "$A$ proves $B$'', and $A\vDash B$, read if $A$ is true then $B$ is true, are referred to as syntactic and semantic consequence, respectively.
In the history of ...
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Who first proved that empty set is subset of all sets?
Who is the mathematician who proved that empty set is subset of all sets and made it known to most mathematicians? I looked into the ripple effects in the mathematical world that would occur if the ...
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When did people first thought of a purely symbolic logic?
In Euclid's Elements, the famous five planar geometry axioms are formulated in common language (ancient greek in this case) and use ambiguous terms.
On the other hand, modern theories like ZFC or ...
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Origin of vacuous truth
I am curious about vacuous truth in logic. I searched Google for various resources, including here. What I could learn from the paper was that I could not know the history or background of vacuous ...
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Did Dedekind's work directly influence the work of Hilbert?
I am wondering if Dedekind's theory about the structure of deductive science influenced the work of Hilbert.
Hilbert obviously favored axioms at the beginnings of a deductive science, whereas Dedekind ...
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Who was the first to define the limit of a convergent sequence with quantifiers $\forall$ and $\exists$?
I mean this definition:
A sequence $(u_n)_{n\in \mathbb{N}}$ converges to a limit $l$ if and only if:
$$\forall \epsilon>0 ~~\exists N \in \mathbb{N} ~~ \forall n \ge N ~~\vert u_n -l \vert < \...
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Etymology or genesis of the concept of proof
I'm wondering about the origin of the concept of proof in human thinking. Nowadays the concept is a cornerstone of mathematics, but also in law. Did early mathematicians maybe adopt it from the ...
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How did someone discover LCM?
How did someone came up with an idea that if we do prime factorization of two numbers and then multiply all the prime factors but including common ones only once, we will get a number that is the ...
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When did mathematicians realize that theory of algebraically closed fields admits quantifier elimination?
A nice property of algebraically closed fields is that the theory that describes them ($ACF$) admits quantifier elimination: any statement can be shown equivalent (in the theory) to another statement ...
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Why did Hilbert believe consistency implies existence?
I am reading Sieg's "Hilbert's programs and beyond" and I am having difficulty understanding this quote by Hilbert on page 74:
In the Paris Lecture Hilbert re-emphasized and expanded this ...
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Who was the first to write proofs in this manner?
Who was the first to write proofs in this fashion?
By ``in this fashion'' I mean, using three columns, which go like:
Line number. Premise or assertion. Justification.
Line number. Premise or ...
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Use of the verb "induct" in proofs by mathematical induction
Occasionally, in a proof by mathematical induction, the writer will say something like, "We induct on $n$" or "We induct on the number of vertices." This usage of the verb induct ...
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Has Penrose ever acknowledged criticism of the Penrose-Lucas argument?
So, Roger Penrose is a bright guy, I mean, he won the Nobel Prize, but the Penrose-Lucas argument that the human mind is a hypercomputer based on Gödel's Second Incompleteness Theorem is laughably bad....
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What did George Boole contribute to order theory?
The Wikipedia page says for order theory says:
As explained before, orders are ubiquitous in mathematics. However, earliest explicit mentionings of partial orders are probably to be found not before ...
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Development of Logic: Intuitive use vs what took so long to formalize?
I was encouraged in MathOverflow to ask this question here - so here it goes: As I am reading Frege and Boole, I am struck by that there doesn't seem to be a great alternative to Aristotle's Logic ...
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First use of corner quotes for Gödel numbers
Who first used the corner quotes, ⌜ and ⌝, or $\texttt{\Godelnum}$ with Sam Buss's macro, for the notion of Gödel number?
Quine introduced corner quotes, but did not use them for the notion of Gödel ...
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I'm wondering how the Egyptians figured out the false position method in solving their equations?
I'm wondering how the Egyptians figured out the false position method in solving their equations? Is there a good source that can tell me how they thought through this?
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Is C.S Peirce really an independent co-discoverer of the first-order logic?
According to this article copies of Frege's Begriffsschrift were both present during the early 1880s (before Peirce published his works on first-order logic) at the Johns Hopkins University, where ...
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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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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 ...
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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 ...
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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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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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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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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 ...
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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 ...
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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? ...
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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:
...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 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?
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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, Hilbert, and Poincaré formally define or directly discusse 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, Poincaré, and Cantor were ...
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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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