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Questions involving philosophy of mathematics. Please consider if Philosophy Stack Exchange is a better site to post your question.

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17
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4answers
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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 ($\...
253
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24answers
20k views

Is mathematics one big tautology?

Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties ...
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3answers
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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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1answer
123 views

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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2answers
80 views

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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6answers
605 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 ...
5
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3answers
517 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 ...
4
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3answers
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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1answer
21 views

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
47 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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0answers
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. ...
5
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5answers
749 views

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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3answers
137 views

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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1answer
213 views

Did some ultra-finitists suggest which number should be the largest?

I came across the ultra-finitism, the idea that there is a "largest number". Even most ultra-finitists admit that the "largest number" cannot be exactly defined. Therefore my question : Did some ...
37
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15answers
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The problem of instant velocity

The concept of velocity is by definition the movement divided by the interval of time between initial position and final position. If $f(t)$ is the position of a particle at time $t$; the velocity in ...
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3answers
122 views

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 ...
2
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0answers
257 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, ...
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0answers
37 views

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
38 views

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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7answers
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Why do we stop at exponentiation stage in arithmetic of natural numbers?

In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an iteration of $S$. ...
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0answers
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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15answers
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Is there such a thing as proof by example (not counter example)

Is there such a logical thing as proof by example? I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right. This works and is ...
4
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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 ...
3
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3answers
593 views

Logic & Reality [closed]

Maybe just a quick preface first before the question. I recently started a YouTube channel where I'm trying to clear up confusions I see on various (usually philosophical topics). In my 2nd video, the ...
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4answers
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Is McGee's counterexample to Modus Ponens accepted by the mathematical community?

In the mid 1980's Vann McGee proposed a counterexample to Modus Ponens: (a) If a Republicans will win the election, then if Reagan will not win, Anderson will win. (b) A Republican will win the ...
4
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1answer
249 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 ...
36
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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 ...
3
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2answers
1k views

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. ...
3
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3answers
1k views

What exactly does tautology mean?

I'm struggling to understand the actual meaning of tautology. I know how to determine tautology with a truth table. But I don't understand what it actually means, so consider the following reasoning: ...
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6answers
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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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3answers
664 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 ...
37
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1answer
4k views

$e^{e^{e^{79}}}$ and ultrafinitism

I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. ...
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0answers
149 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, ...
0
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1answer
78 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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4answers
764 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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1answer
178 views

Where do I go wrong with Presburger “multiplication”?

This is a speculation about the Presburger arithmetics, the Peano axioms with only addition added, vis-à-vis the same axioms with both addition and multiplication added. In the first case I understand ...
18
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3answers
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 ...
9
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3answers
394 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
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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. ...
1
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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 ...
460
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36answers
58k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
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4answers
2k views

Good Sources for Lecture Movies in Set Theory, Logic and Philosophy of Maths

Of course as any other researcher I'm not able to attend any scientific event in my research area. But it is always interesting and useful to watch the lecture movies of these events. I will ...
4
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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 ...
4
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1answer
636 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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1answer
1k views

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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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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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: ...
3
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3answers
491 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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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 ...
2
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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 ...