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Do only certain people exceed at math well? [closed]

It's obvious if you look around that math has always been one of the toughest subjects in all areas, from federal-traditional public schools to simply people learning it as an autodidact, hobby, or as ...
2
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1answer
137 views

Difference between impredicative and predicative version of separation axiom

What is the difference between an impredicative and a predicative version of the separation axiom in ZFC: $$\forall x \exists y \forall z ( z\in y \leftrightarrow (z \in x \wedge \phi (z)) $$ What ...
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2answers
135 views

Are axioms chosen with the goal of “making things work” instead of some deep philosophies?

Are axioms chosen with the goal of "making things work" instead of some deep philosophies? If everything should be deducible, that is, provable from something else, then in this chain of deduction ...
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10answers
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How can Zeno's dichotomy paradox be disproved using mathematics?

A brief description of the paradox taken from Wikipedia: Suppose Sam wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must ...
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2answers
162 views

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works.

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works. I think Bertrand Russell was a special mind and I set a goal for myself to study all his ...
3
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3answers
83 views

Logical 0, binary 0, decimal 0: are they the same?

Logical 0, binary 0, decimal 0: are they all the same in mathematics? A programming language might treat them differently, but is 0 just 0? No matter whether it is logical, binary, decimal, ...
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3answers
182 views

Disturbing the foundations of mathematics

I was curious of knowing if it is possible that an event "x" could disturb so greatly mathematics that we could be casting doubts on all the achieved results from the very beginning. I'm not sure if ...
2
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1answer
170 views

Why should we accept the existence of subsets $A$ such that neither $A$ nor $A^c$ are recursively ennumerable? And how can we persuade others?

Encode every pair $(t,x)$ (where $t$ is a Turing machine and $x$ is an input string) as a distinct natural number. Then the halting subset $H$ fails to be recursive. $$H := \{(t,x) \in \mathbb{N} \...
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17answers
5k views

Is a proof still valid if only the writer understands it? [closed]

Say that there is some conjecture that someone has just proved. Let's assume that this proof is correct--that it is based on deductive reasoning and reaches the desired conclusion. However, if he/...
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1answer
91 views

When is it useful to reduce mathematical objects to foundational levels and when it is not?

When is it useful to reduce mathematical objects to foundational levels and when it is not? Let's say you work in the field of computer vision, or else. How can you claim your method is optimal if ...
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3answers
278 views

Are there other approaches for the foundations of mathematics, other than logic and set theory?

Are there other approaches for the foundations of mathematics, other than logic and set theory? And why does set theory begin talking about objects and groups of objects. Is it proven somewhere that ...
0
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1answer
117 views

If a set is a group of objects, then what is an object? [closed]

If a set is a group of objects, then what is an object? My best try at this is the following: An object is anything that we can discuss or think about, separately from everything else. It is not ...
2
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1answer
110 views

Question about the univalence axiom versus skeleta.

Here, Dan Licata writes: [Univalence] can be used to build algebraic structures in such a way that isomorphic structures are equal (e.g. equality of groups is group isomorphism). He writes ...
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1answer
69 views

Constructivist Interpretation of a Function

Lets suppose I have an exponential function $a^{x}$, and I desire to show that for any number $n$ in $(0, \infty)$, it is possible to find a value of $x_0$ such that $a^{x_0} = n$. The simplest proof ...
2
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2answers
906 views

Consistency of ZFC and the key assumption [closed]

I recently read this answer to a MathOverflow question that got me thinking. Very roughly, here's what the author of that answer says: Gödel's second incompleteness theorem implies that if there ...
2
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1answer
120 views

Do circles exist

So I was wondering about circles today and if they really do exsist. If you graph a circle in function mode, your equation looks like$$y=\sqrt{1-x^2}$$ Now for simple purposes lets take a portion of ...
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2answers
309 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in x~~~S(y)\...
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1answer
67 views

Variable theory

I wanted to know if there's any alternative variable theory to Russell's used in his Principia. I mean, the modern definition of "variable" and "constant" still follows his works? I try to search on ...
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3answers
325 views

Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?

It is frequently stated (in textbooks, on Wikipedia) that the "Law of large numbers" in mathematical probability theory is a statement about relative frequencies of occurrence of an event in a finite ...
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6answers
339 views

Logical issues with the weak law of large numbers and its interpretation

In several probability textbooks I have found what amounts to the following argument: Let A be an event in some probabilistic experiment. Let p=P(A) be the probability of this event occurring in ...
2
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2answers
153 views

How an axiomatic system is made?

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call ...
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4answers
162 views

“The enrapturing discoveries of our field systematically conceal (…) the analogical train of thought that is the authentic life of mathematics…”

In the preface of the book Discrete Thoughts, Gian-Carlo Rota writes: Sometime, in a future that is knocking at our door, we shall have to retrain ourselves or our children to properly tell the ...
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2answers
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How does undecidability of 'theoremhood' imply that human ingenuity is necessary in mathematics?

In Robert Stoll's "Set Theory and Logic", there is the following passage on effectiveness of theorems (p. 375) : Mathematical logicians have shown that for many interesting axiomatic theories ...
3
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1answer
186 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
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0answers
85 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
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1answer
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First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second ...
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1answer
75 views

What exactly does $\vdash_T G_T \leftrightarrow \lnot \exists y$ Prf$(\ulcorner G_T \urcorner, y)$ mean?

To me this translates to: $G_T$ is provable in $T$ if and only if there doesn't exist a $y$ such that $y$ is a witness to the provability of $\ulcorner G_T \urcorner$. But I'm not entirely sure what ...
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3answers
323 views

Is math independent of our sensory experience? [closed]

I've been asking myself this and other questions in the field of philosophy of mathematics. Could we, if we were isolated from any kind of sensory experience, be able to learn mathematics? Also, what ...
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4answers
210 views

Where does the importance of math come from? [closed]

It is a somewhat philosophical question. I personally believe that the importance of math is due to its usefulness and lots of applications. Mathematics is used in everywhere nowadays; as Ian Stewart ...
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1answer
87 views

Sheafs appearing in philosophy?

I apologize in advance if I make mistakes in the following construction. I have very recently been introduced to the concept of a sheaf. I am currently a mathematics major and philosophy minor and ...
2
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1answer
74 views

I currently know Calculus I — What steps would I take to understand Zermelo–Fraenkel set theory?

While this question can be discussed, it should have a clear answer by stating the following: How can one go from a high school / low-level college understanding of mathematics (completed Calculus 1)...
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3answers
118 views

Regarding the validity of probability theory [closed]

Imagine I have a regular balanced dice and i roll it once. It is assumed that the probability of any number (1-6) is 1/6. However, isn't this just an illusion we are feeding ourselves for our lack of ...
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25answers
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Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...
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1answer
65 views

What exactly is a property?

How is a property $P$ formally defined in mathematics? I mean for example if $f$ is a morphism from an object $X$ to $Y$ in some category, then somehow I feel that "has codomain $Y$" is too broad to ...
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1answer
75 views

Need help locating a paper

One of the references of the paper Paulo Régis C. Ruffino, A Criticism on "A Mathematician's Apology" by G. H. Hardy (arXiv:1112.4499 [math.HO]) is: Vershik, A. M. – A Dangerous Joke, The ...
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3answers
215 views

Can I create my own function like Trigonometric or Exponential

When I want to solve mathematical problems, most of the time I meet the following functions Algebraic like polynomials. Trigonometric like sin(), cos(), tan(), cot(). Logarithmic like log(). ...
3
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1answer
149 views

Theorems that we can prove only by contradiction

While most of the world is fine with proofs performed by contradicting the thesis, direct proofs are sometimes considered more elegant than indirect ones. Those who prefer intuitionism or ...
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4answers
264 views

Soft question: Examples where implications derived from mathematical models failed to describe reality

I have always been fascinated by how well conclusions drawn from mathematical models could fit reality, so I wondered if there are any counter examples. In "Gödel, Escher, Bach" I could already find ...
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264 views

How to explain ✳43.3 and ✳43.31 in Whitehead and Russell's PM?

Take ✳43.3 for example, I presume $ P = R |Q $ where R is fixed. $ R| $ is the relation between $R|Q$ and $Q$, ie. $ R| = \hat{P} \hat{Q} \{ P = R|Q \} $ $Ɑ‘R|= \hat{Q}\{ E! R|‘Q \}$ Given that $...
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0answers
86 views

A Question Regarding Ordinal Turing Machines

Consider the following theorem of Koepke: 'A set x of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of the constructible universe L". Taking ...
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3answers
439 views

Visualizing mathematics and geometry

Im writing a paper on the role of visualization in mathematics and specifically geometry. I was wondering if it is possible to represent any arbitrary system of relations and manipulable objects ...
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2answers
479 views

Exactly who popularized the modern definition of domain and codomain of functions?

In Whitehead and Russell's Principia, domain is the referents of relation; converse domain is the relata. Modern function in mathematics is just one special case of relation whose referent is unique ...
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242 views

Implications and Ordinary language

I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources. One particular case is sentences in the form $...
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4answers
244 views

Is mathematics a science? [duplicate]

Is mathematics a science? I have long considered this to be open to debate, but my professor said that he once heard the quote, ...
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2answers
949 views

Who stole the axioms in Natural Deduction?

The study of Gentzen's sequent calculus give me the opportunity to propose some reflections about the concept of logical truth. I'll refer to the english edition of Gentzen's works : The collected ...
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2answers
110 views

About the concept of logical truth

From Frege and Russell to modern mathematical logic textbooks, there were a "shift" of focus from the concept of logical truth, through that of valid formula, to the current concepts of logical ...
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324 views

Gödel's Completeness Theorem and logical consequence [closed]

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
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2answers
123 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
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1answer
156 views

In an infinity of choices, is it possible to guess the correct one?

So I've been thinking about the infinite universes model, where each possible action or event creates a new universe for each outcome. For example, if you flip a coin there will be one universe in ...
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1answer
306 views

Large Operators?

Large operators have always seems strange to me, sometimes their meaning is based on the symbol and other times it has no correlation. For instance, the summation (sigma) has no relation to the ...