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Link to a paper

I remember seeing, in one of the comments to one of the top-voted questions here, or on MO, a link to a paper. It contained, in addition to other things, a part about the dangers of doing mathematics....
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Why did mathematician construct extended real number systems, $\mathbb R \cup\{+\infty,-\infty\}$?

I know some properties cannot be defined with the real number system such as supremum of an unbounded set. but I want to know the philosophy behind this construction (extended real number system ($\...
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Since model theory must be founded in set theory, do you get different model theoretic results depending on your choice of set theory?

As I understand it, model theory tells you a lot about set theories (plural) but is also founded upon set theory, which means you need to pick one set theory to do model theory within. But different ...
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How best to mathematically describe 'Tristram Shandy problem'

This question is related to a question posted on philosophy.SE, but I am posting it here since it is mathematical in nature, and since answers using LaTex are available here. (I will be providing an ...
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Is mathematics just a bunch of nested empty sets?

When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying ...
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3answers
519 views

Why do finitists reject the axiom of infinity? [closed]

The axiom of infinity implies that there exist infinite sets. We can construct the natural numbers without this axiom, but we cannot put them together in a set, as this would violate this axiom. The ...
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356 views

Skolem's paradox showing us that we might be trapped in our view of the world

According to Skolem's Paradox, ZFC as a first order axiomatization of set theory has a countable model, but allows a proof that uncountable sets exist in every model of ZFC. It becomes counter-...
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Statistics What is the Standard Error?

Bookstores like education, because national data show that $71$% of college graduates have read a book in the past year, compared to $54$% of the general population age 18 and over. The data also show ...
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1answer
48 views

Philosophy Statistics Standard Deviation

Consider the following list: (A) 1, 3, 4, 5, 7. What is the standard deviation of list A? Is this data being asked from a population or a sample? I think it is being asked from a sample, and that ...
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51 views

philosophy statistics identity null hypothesis

Many companies are experimenting with “flex-time,” allowing employees to choose their schedules with broad limits set by management. Among other things, flex-time is supposed to reduce absenteeism. ...
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Set theoretic concepts in first order logic

I have been reading introductory texts on first order logic (for example, Leary&Kristiansen). All of them used concepts that I have heard in set theory courses - ordered pairs, functions, ...
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Friday analysis of the unexpected hanging paradox [closed]

The judge told me: A1. You will be hanged on day X. (X is some day from Monday to Friday) B1. You can't deduce what X is. It's Friday morning and I'm still alive. My first deduction is (please tell ...
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Circleness of a Circle [closed]

Can it be rigorously proven that a circle is the circle? It is assumed that the path traced by a revolution of a segment about a point is the one and only shape of a circle, but can any other shape ...
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Question about set theory and first order logic [duplicate]

I was studying first order logic by a book and the author uses set theory to define some concepts like models, structures, etc. Everything fine here, but then i started to read about ZFC set theory ...
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0answers
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Which of the three options would you stumble upon if you tried to prove that the Münchhausen trilemma is real/true [closed]

In the wiki page of the Münchhausen trilemma I read that nothing can really be proved. Why should I then take this information seriously? How would you go about proving that this trilemma is a real ...
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Worrying about logic and foundations too much [closed]

I am studying undergraduate physics and I am always interested in why we consider certain physical models instead of the others. That ultimately leads to the question why we consider one mathematical ...
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3answers
1k views

Is “This sentence is true” true or false (or both); is it a proposition?

From what I understand, a proposition is either true or false, but not both. "This sentence is false" can be neither true nor false and is thus not a proposition. However, is "This sentence is true" ...
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144 views

Godel's Theorems and Conventionalism

Philosophers sometimes ask which axioms are "true". Of course, no one believes that there is a serious question as to whether the axiom of commutativity for groups, or the Parallel Postulate, is true....
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328 views

What's the difference between a proof and a derivation?

I did a BSc in Theoretical Physics, meaning that a lot of my time was spent deriving equations, making hand-wavy arguments, and arriving at solutions with a distinct lack of rigour. I'm now doing an ...
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3answers
668 views

Why is negative minus negative not negative? Why is negative times positive not directionless?

What I understand is that the only difference between plus and minus is direction. I've never understood this: That +1 + +1 is ...
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150 views

Why linear congruential generator is called random number generator?

As far my understanding linear congruential generator is used to produce pseudo random number. But the algorithm requires four parameters like, ...
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1answer
79 views

Is it possible to construct a formal system such that all interesting statements from ZFC can be proven within the system?

I'm familiar with Godel's incompleteness theorem, which very basically states that there exists a statement that can be neither proved or disproved within a formal system powerful enough to include ...
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765 views

Decidability and “truth value”

One can read in the Wikipedia page for "Gödel's incompleteness theorems": Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of ...
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2answers
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Difference between provability and truth of Goodstein's theorem

I have been thinking about the difference between provability and truth and think this example can illustrate what I have been wondering about: We know that Goodstein's theorem (G) is unprovable in ...
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3answers
136 views

Should axioms be seen as “building blocks of definitions”?

This question is about the difference between a definition and an axiom. However, it does not address the following point: Whenever we define something, this is often written as a series of axioms. ...
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1answer
109 views

Does Planck length contradict math? [closed]

I have a general question about math and infinity which really bothers me as a math student - can we actually divide every length by two? I would like to believe the answer is yes, because it ...
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1answer
186 views

Bourbaki influence on the current mathematical research

I know that the bourbakism had an influence in the academic world and an impact in secondary school and university. There are remarks in this Wikipedia dedicated to Nicolas Bourbaki. Question. Does ...
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2answers
354 views

Is mathematics aprioristic? [closed]

Is mathematics aprioristic? I do not know. Some axioms of arithmetic and geometry arose clearly inspired by the observation of Nature. After that, those areas of mathematics were often developed with ...
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1answer
646 views

Why did we settle for ZFC?

I understand the need for a solid foundation of mathematics. But it seems to me that we have definitely settled for ZFC and I would like to know if there are strong reasons for this or not. I am not ...
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2answers
68 views

Is it legitimate to use methods of one mathematical branch in other branch of mathematics ? [closed]

For example, is it legitimate to use methods of mathematical analysis in number theory ?
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228 views

The importance of prime numbers in physical theories

Imagine that a friend asks me about what is the importance of prime numbers in physics. What should I tell him/her? I know that natural numbers should be important in quantum mechanics because there ...
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2answers
126 views

What happens to a point when you rotate a line?

If I have a line on an $XY$ grid: o---o---o---o---o a b c d e And I rotate it like so: ...
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3answers
128 views

Good resources on the intersection of probability theory and logic from a foundations/philosophical perspective?

What are some good books, courses, or online resources for probability theory that highlights differences between classical, frequentist, Bayesian, epistemic etc.? I majored in philosophy and am now ...
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3answers
151 views

Are the laws of mathematics 'absolute' in this universe? [closed]

We observe that almost all physical phenomena (which has been explained) can eventually be explained by the laws of mathematics. Mathematics seems ubiquitous- for example the form of the differential ...
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48 views

Are there examples of theorems incompatible with ZF that have proven useful in the sciences?

I was thinking about "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and I realized all the mathematics I know about in science are compatible with ZF (even if they sometimes ...
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Is 0 the absence of something? [closed]

In computer programming, null and 0 values are two different things. I was wondering if the same applied to just mathematics in general. The reason I ask this question is because I don't understand ...
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1answer
93 views

'The Computer and the Brain' - The mathematical language of the brain

I couldn't decide whether this question is more appropriate to post here or at the philosophy SE, but I thought I'd give people with a mathematical perspective the opportunity to help me decide. I'm ...
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1answer
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Why and how Mathematicians define unity?

I hope this one (pun intended) post won't get ripped by the community. I wondered what are the most abstract ways to define unity element? Why is there a need for unity element in general? Is it just ...
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Does advanced math “power” more rudimentary math?

I came across this quote by Eric Weinstein that "when things got supposedly more advanced, they actually got simpler because mathematicians started revealing what was powering all the things that you ...
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1answer
65 views

Supposing that numbers are reducible to sets, how would one go about reducing a complex number to one?

I'm aware of the Benacerraf's identification problem, but suppose that numbers are reducible to sets such that the empty set is identified with zero, the power set of the empty set is identifiable ...
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1answer
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Are category-theory and set-theory on the equal foundational footing?

Set-theory is widely taken to be foundational to the rest of mathematics. So is category-theory. My question is: Are they two alternative, rival candidates for the role of a foundational theory of ...
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1answer
53 views

Logarithm, exponentiation, addition, and multiplication,

If the logarithm converts multiplication to addition, thus simplifying mathematics, does exponentiation, converting addition to multiplication, "complicate" mathematics? I've only ever seen arguments ...
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2answers
373 views

Axioms in Gödel's ontological proof are inconsistent?

So, my problem is with Axiom 5 of the proof, where Gödel considers necessary existence as a property. However, by his own definition, a 'property' applies to objects including those whose necessary ...
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777 views

The boundaries of mathematics?

There are five possible answers to a multiple-choice question. Given that the student does not know the answer, what is the probability that the student chooses the first answer? That is not a well ...
4
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3answers
290 views

Are numbers real, in a metaphysical sense? [closed]

I live and work with numbers almost all the time and have done so for most of my 77 years. I can almost feel them. But it is only almost. I have to believe, contra Plato, that they and moreover all of ...
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2answers
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What kind of alternative mathematics systems exist? [closed]

What kind of alternative mathematics systems exist? What I mean is, mathematical systems that use a different sort of "basic premises" or e.g. logic(s) than the contemporary "mainstream" mathematics. ...
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2answers
210 views

Axiomatic formal language theory

I am struggling with this problem of philosophy of mathematics, that is to apply the axiomatic method to give foundations to the theory of formal languages, without mention explicitly the concepts of "...
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0answers
90 views

How can one think conceptually about Type Theory when one explains the differences between ZFC and Type Theory?

It's a big question, this I am quite aware of, so please excuse my little understanding on the subject but with reference to the following question, to which I would rather like to extend a little, ...
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2answers
100 views

Russell's “On Denoting,” formalization

So, I was reading Russell's paper 'On Denoting' and stumbled upon the (in)famous paraphrase\analysis of the definite description "x was the father of Charles II." As it is known, Russell's paraphrase ...
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0answers
258 views

Can we hope for an elementary proof of a conjecture of Goldbach?

It is written on Wikipedia: "During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, ...