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why learning math in elementary school was harder for me rather than upper grades? [closed]

When I was an elementary student, I'd suffered from understanding basic things like multiplication table and other simple things and I had to memorized them. Last hours I was searching for genesis of ...
User14373's user avatar
2 votes
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130 views

When mathematicians say "true" do they mean "true in all models"?

According to the comments to this question, Truth is ordinarily defined by reference to models. If so, even axioms and theorems are not true without reference to a model. However, when ...
MathMan's user avatar
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Is every mathematical statement provable?

For my purposes, I'm defining provable as "there exists a rigorous and formal proof that the statement is true or false". In my mind, at first, it seemed obvious that if a statement must ...
Howard Pyle's user avatar
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Relationship between Philosophy and Mathematics [closed]

Many famous mathematicians were philosophers. But now philosophy and mathematics are different fields and a person who learning one do not know much about the other. Actually, I have studied both and ...
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In real life, we can have a pencil of length 2 cm. Can we have pencil of length $\sqrt{2}$ cm? [migrated]

In real life, we can have a pencil of length 2 cm. Can we have pencil of length $\sqrt{2}$ cm? My answer to that was no , we cannot even make 2 cm pencil. My argument was that when are working ...
user157835's user avatar
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Arithmetization of Turing machines

Refer to Turing's 1936 paper, page 248, last paragraph. I present the paragraph in verbatim below : The expression "there is a general process for determining..." has been used throughout ...
Ajax's user avatar
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Benefits/uses of non-base 10 number systems?

For reference, I'm studying math and anthropology at university, and I've been dying to find some overlap of math theory and ethnomathematics (math uses/tools/systems/etc in other cultures). I'm ...
Rhinestone's user avatar
6 votes
2 answers
770 views

Formally how do we view finite sets

This might be silly, but I have been thinking about how we would work with finite sets very formally. So, $\{1,2,3,\cdots,n\} = \{k \in \mathbb{Z}^+ \mid k \leq n\}$ gives a representee for which any ...
MathNerd23571113's user avatar
3 votes
2 answers
248 views

What are fun mathematical facts for non-mathematicians? [duplicate]

I like to spend my life with mathematics. I think it is the best thing I can do in my life. However, I have great difficulty explaining what I am doing to non-mathematicians, even educated ones. For ...
boyler's user avatar
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Why is it important to prove that some particular set is a vector space as opposed to just asserting such objects exist?

In Axler's Linear Algebra Done Right Example 1.24, we are asked to prove that the set of all functions from some set S to the set of real (or complex) numbers is a vector space. I proved this by using ...
Antarctica07's user avatar
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How to get LNC as a theorem using Frege's Prop Calculus?

So Im using axioms from,Frege propositional calculus and is there any way to derive Law of non contradiction as theorem from them. The axioms A → (B → A) | THEN-1 (A → (B → C)) → ((A → B) → (A → C)) ...
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What are the differences between equality, equations and identities? [duplicate]

What are the differences between the followings: Identity $$ \sin^2(\alpha) + \cos^2(\alpha) = 1 $$ Equation $$ 4x = 16 $$ Equality - $x,y$ are mathematical objects. $$ x = y $$ All of the three ...
mawaior's user avatar
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2 answers
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Is there a mathematical or physical, real world use for numbers passed I? Who determines what can be conceptualized or not? [closed]

I is the Square root of -1, such that I * I = -1. Through this, I can be considered like a "Half Negative." Why hasn't this been taken further? Why don't we make a quantity such that I^...
Kyotiq's user avatar
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Set theory and model theory: which set is ZFC?

I have yet another post about what is model theory doing and why is it valid; I hope I can be coherent. (1) https://mathoverflow.net/questions/23060/set-theory-and-model-theory (2) What exactly is the ...
Riley Moriss's user avatar
13 votes
2 answers
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Probability - Interview Question - Hidden Assumptions and Phrasing Issues

I’ve encountered the following seemingly simple probability interview question in my workplace: Two reviewers were tasked with finding errors in a book. The first had found 40 errors and the other ...
Yonatan Harari's user avatar
2 votes
4 answers
362 views

How to interpret what a set is to see how it could be infinite?

Currently, 'infinite set' sounds oxymoronic to me, so my question is how to interpret what a set is such that it is consonant with it being infinite. I understand that we take it as axiomatic that ...
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Are opinions considered sentences in Logic?

I am beginning to read the book titled forallx An Introduction to Formal Logic by P.D. Magnus. This is an open source book. On page 4 Magnus states: In this open source book found here: https://math....
Dr. J's user avatar
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Is there a mathematical notion of "why"?

Is there a mathematical notion of "why"? That is, are there reasons behind the truth of certain mathematical statements? Personally, my belief is that true mathematical statements just are ...
user107952's user avatar
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I don’t know what a natural number actually is, and it’s making me sad :( [closed]

For some context I did a course on set theory where I was taught about ZFC, and the construction of the natural numbers, integers etc. I think I was far too young to take the course because it’s left ...
Fraser Pye's user avatar
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1 answer
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Why do constructive mathematicians claim that mathematical truth is temporal?

It seems to me (and correct me if this is a misconception) that the traditional divide in the interpretation and practice of mathematics is between platonists, who believe that mathematical objects ...
user9812063's user avatar
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1 answer
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What is the deal with the bizarre philosophy in historical and current axiomatic set theory? [closed]

Many respectable mathematicians have written about "true axioms" or similar concerns about whether all mathematical theorems are in fact "real" or "true". This seems to ...
May Emerson's user avatar
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Is there any logic system which ENTIRELY rejects non-contradiction of any kind for any sentence (i.e. all contradictions are true)? Is this possible?

I've recently learned about paraconsistent and intuitionistic logic, and dialetheism. According to the Stanford Encyclopedia of Philosophy's page on Dialetheism, it states: Dialetheism is the view ...
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What makes one proof different from another one? [duplicate]

There are around 370 different ways to prove the Pythagorean Theorem, but what does that exactly mean? For instance, if your proof states that $x^2+y^2=z^2$, I could construct a different one by ...
nuuusxd's user avatar
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Why is list of names no more capable of expressing a proposition?

From the Open Logic Project book 2.2, Philosophical reflections (Set theory): Third: when we “identify” relations with sets, we said that we would allow ourselves to write Rxy for ⟨x, y⟩ ∈ R. This is ...
solvable group's user avatar
2 votes
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Does it make sense to divide mathematical theorems into those susceptible to experimental conjecture and those that are not?

I read the following in a good book: ”let's start from zero", the authors are Vinicio Villani and Maurizio Berni (pisan mathematicians) but I don't know if the book is also marketed outside Italy....
Fausto Vezzaro's user avatar
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1 answer
204 views

What is the purpose of mathematical research? [closed]

This is a bit of a soft and philosophical question, but what is the purpose of mathematical research? It seems to me that there is no end goal of mathematical research, because everything can be ...
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Why do we have different sets of axioms? (metamathematics reference request)

For example, ZFC and ZF. I have come across the notion of pure and applied mathematics, and how the development of the former can (and is usually intended to) lead to the furtherance of the latter. In ...
jacob2222's user avatar
1 vote
1 answer
252 views

What exactly are capture and release?

Motivation: I'm interested in how different people resolve the Liar paradox and other, related phenomena, like the revenge Liar paradoxes, and so on. I have a copy of "Formal Theories of Truth,&...
Shaun's user avatar
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(Soft Question) Real World Modeling as Understood Through Pure Math

This question is necessarily vague; I'm not looking for an answer so much as I'm checking to see if this is something that has been thought about/discussed before, and if there are any resources out ...
Nick A.'s user avatar
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in modal set theory, why it is issue?

I have been studying the Iterative Set Concept within the context of the paper titled "Modal Set Theory" from Menzel specifically on pages 11-12. "As we’ve just seen, the iterative ...
유준상's user avatar
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1 answer
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Why do numbers apply to such disparate concepts?

I understand numbers to be defined as objects defined to have certain convenient properties in relation to certain operations. It is very surprising that the exact same group objects should be ...
tom894's user avatar
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Is there a modal operator, L, that satisfies φ ↔ 𝛙 & ~L𝛙 ⊢ ~Lφ?

I am wondering if there's some modal operator that would satisfy $$φ ↔ 𝛙 \& ~L𝛙 ⊢ ~Lφ.$$ That is: Given $φ ↔ 𝛙 \& ~L𝛙$ You can get to $~Lφ$ One limitation is that $L$ for sure does not ...
Melody's user avatar
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7 votes
1 answer
289 views

How should one understand the "universe of sets"?

One way to understand the axioms of $\mathsf{ZFC}$ is to see them as a describing the "universe of sets" $V$, together with the "true membership relation" $\in$. The universe $V$ ...
Joe's user avatar
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2 votes
0 answers
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Why were evil numbers named "evil"?

According to the entry for Sloane sequence A001969: Evil numbers: nonnegative integers with an even number of 1's in their binary expansion. And deeper inside that entry it states that The terms &...
Fomalhaut's user avatar
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0 votes
1 answer
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Why do the increasing sequences $x_n, y_n$, decrease to $x,y$ to show a bivariate (or univariate) cdf is right continuous?

I have a problem understanding why the concept of right continuity of a cdf has to decrease a sequence $x_n$ or $y_n$ to a limiting value $x$ or $y$ respectively. I do not understand why in this ...
TopoSet32's user avatar
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0 answers
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Is expression and its result is the same thing?

So, $\frac{1}{2}$ and $0.5$ are just two different ways to address the same object which is rational number $\frac{1}{2}$. What about more complex expressions? Like $\{a, b\} \cap \{b, c\}$ is just ...
tbsd's user avatar
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On series expansions and valid arguments

Hopefully the answer to this question isn't so obvious one way or the other that it ends up just creating more confusion. Briefly, I'm concerned about the potential for subtly illegal moves to creep ...
RTF's user avatar
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2 answers
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Is the deducutive apparatus of a formal system necessarily a set of inference rules?

In the book "Logic" by Paul Tomassi, the author uses the term deductive apparatus to refer to the set of inference rules in propositional logic and first-order logic. The use of this term ...
RyRy the Fly Guy's user avatar
5 votes
4 answers
1k views

Is proof of the law of identity a case of circular reasoning?

I am reading "Logic" by Paul Tomassi. While discussing first-order logic in Chapter $6$ p. $310$, he provides the following justification for the inference rule known as identity ...
RyRy the Fly Guy's user avatar
3 votes
4 answers
732 views

what is the exact meaning of the identity relation? [closed]

I would like some clarification regarding the exact meaning of the identity relation. Specifically, if $a=b$, does this mean $(1)$ $a$ is the very same object as $b$ or does it leave open the ...
RyRy the Fly Guy's user avatar
4 votes
1 answer
75 views

How do proofs of program termination depend on strength of logical systems?

I'm looking for clarifying insights on the following topic. While there can be no general proof strategy to show that terminating Turing programs do, indeed, terminate, some specific programs can be ...
cxdorn's user avatar
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1 vote
2 answers
235 views

On the Consistency of Non-Euclidean Geometry

I've recently found a really old "Philosophy of Math" book in my University library, and in the book it says that the it has been proven that : if non-euclidean geometry is inconsistent, ...
Diana's user avatar
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-1 votes
2 answers
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Presupposition in logic [closed]

Let S be a presupposition for Q. For example: S: I used to smoke Q: I quit smoking As far as I know, Q has a truth value if and only if S is true. However, my understanding of the concept of ...
Егор Галыкин's user avatar
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0 answers
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Using set-builder notation in an unusual way for more concise expressions in a definition. Good style?

What do you think about the following usage of the set-builder notation $\{ x \mid P(x) \}$? Let $Y = \{ p_1, \ldots, p_n \}$ be $n$ objects and $A,B$ subsets of $Y$. Define a function from $\{1,\...
StefanH's user avatar
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2 votes
0 answers
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Did Skolem (and others) consider all "legitimate" models to be "actually" countable?

In Thoralf Skolem's Remarks on Axiomatized Set Theory (van Heijenoort translation), Skolem says: There is no contradiction at all if a set $M$ of the domain $B$ is nondenumerable in the sense of the ...
NikS's user avatar
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0 votes
1 answer
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Different Proof Systems for the same Logic

Some standard kinds of proof systems include natural deductive systems, sequent systems, axiomatic systems, and so on. There are various approaches to each of these kinds of proof systems. The rules/...
PW_246's user avatar
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0 votes
1 answer
116 views

Poincaré's notion of magnitude

This is what Henri Poincaré says in his book Science and Hypothesis (p. 27): Measurable Magnitude.—So far we have not spoken of the measure of magnitudes; we can tell if any one of them is greater ...
Ahmed Abdullah's user avatar
16 votes
13 answers
3k views

Does the epsilon-delta definition of limits truly capture our intuitive understanding of limits?

I've been delving into the concept of limits and the Epsilon-Delta definition. The most basic definition, as I understand it, states that for every real number $\epsilon \gt 0$, there exists a real ...
thomas graceman's user avatar
2 votes
1 answer
279 views

Why are we confident in the ability of ZFC to formalise mathematics if very few proofs are actually converted into ZFC?

$\mathsf{ZFC}$ is often introduced in logic textbooks as a first-order theory with equality and a single non-logical symbol $\in$. However, even stating the axioms of $\mathsf{ZFC}$ in this language ...
Joe's user avatar
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4 votes
1 answer
138 views

Introductory text on logic for those interested in the intersection of logic, algebra, and topology?

I'm a current MA student doing research in formal semantics, which is an application of, among other things, logic and model theory to the study of the semantics of natural languages. I'd like to ...
m. lekk's user avatar
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