# Tagged Questions

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

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### What would qualify as a valid reason to believe there is a closed form?

I noticed that almost every non-homework-level integral posted on this site prompts somebody to ask "Do you have any reason to believe there is a closed form?" (some recent examples here and here) I ...
1k views

### How much math does one need to know to do philosophy of math?

I'm looking for advice from mathematicians who also study philosophy of math (PoM). Due to interest I'd like to study PoM as a hobby, but I'm worried if I don't understand math well enough from a pure ...
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### Is there a proper term and/or symbol for an “agnostic” conclusion?

My question stems from the material conditional: $p \rightarrow q\\p\\\therefore\space q$ However, if $\bar p$ then the conditional is silent. I would like a way to represent this fact using, if ...
9k views

### Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, ... One of my students just rose ...
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### Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book). Since the set of the finite books is countable then the set of real numbers ...
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### Dogmas and Mathematics

What are the dogmas that restrict or promote the development of mathematics? I know that a dogma is a set of beliefs that is accepted by the members of a group without being questioned or doubted. ...
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### Why doesn't $1/x=0$ have any solution?

Just out for curiosity ! Why $1/x=0$ doesn't have any solution? Or is it that the solution takes you to $1=0$ situation which would nullify mathematical principle that we stood for years Educate me....
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### Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
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### looking for good book on the history of formalism

In 1868 Beltrami published a paper ""Saggio di interpretazione della geometria non-euclidea" that seems to have led to the formalist philosophy of mathematics. But what was written exactly what were ...
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### Analogue of prime numbers in addition? [closed]

What is the analogue of prime numbers in addition?
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### True and provably true sentences in a model. Are they the same thing?

In logic, it is said that each sentence in a (consistent) theory is either true or false in a given model. Checking the truth of a sentence in a finite model amounts essentially to finite enumeration ...
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### Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it (...
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### how can we write abstract algorithms?

Writing pseudo-code for algorithms is common practice in the applied mathematics literature. It is also often the case that the ideal input of an algorithm is an infinite set, for example it could be ...
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### Why Is It Rational to Bet on the Most Probable Event?!

Suppose that someone is going to bet in a game. A dice is rolled, and there are only these two options for betting: Option 1. Give 1 dollar and bet on 6. Option 2. Give 1 dollar and bet on 1, 2, 3, ...
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### What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
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### Is there a geometrical proof of the impossibility of squaring the circle?

The impossibility of certain constructions in Euclidean geometry, such as squaring the circle with straight-edge and compass is usually shown by using algebraic methods. I am wondering if there are "...
964 views

### What is a number?

A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and ...
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### Why is CH true if it cannot be proved?

Continuum hypothesis (CH) states that there can be no set whose cardinality is strictly between that of integers and real numbers. Godel, 1940 and Paul Cohen,1963 showed that CH can neither be proved ...
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### Recommendation request: Reasoning behind statistics

It seems, to me at least, that most Statistics textbooks focus on the Statistical methods and techniques, or on the mathematics behind them. Would you recommend me some textbooks (or any online ...
153 views

### Optimal Solution in Natural Deduction

Does there exist an optimal solution for derivations in natural deduction, which is to say that the derivation in question requires the least amount of steps to arrive at the desired conclusion?
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### How much are mathematics driven by applications?

At some point this provocative question came to my mind: Are mathematics mostly driven by applications? I am taking into account some of the comments made to my original question so I want to ...
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### Is it possible to alternate the law of mathematics?

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of ...
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### Are there any natural measures that can be put on the space of models?

I've already asked this question is Philosophy.SE and is a 'soft' question. An undecidable proposition, as in Gödel's Incompleteness theroem, is one whose truth value cannot be determined, because it ...
790 views

### Is there any mathematical meaning in this set-theoretical joke?

Recently I heard a joke: If an object exists, mathematicians call it a set and study it. But if an object does not exist, mathematicians call it a proper class and study it anyway. I wonder, ...
195 views

### Can there be two different math?

As per usual, let PA denote Peano Arithmetic and ZFC denote Zermelo-Fraenkel set theory with choice. Furthermore, ZFC 'validates' PA, in the sense that it proves that the PA axioms hold for the ...
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I've just approached modal logic reading "An Introduction to Non-Classical Logic" of Graham Priest. I am looking for some books that treat this argument in a more extensive way than the book I am ...
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### Books on logic, proof theory and set theory?

I graduated in Computer Science at University of Bologna in Italy some years ago. For various reasons now I am discovering a back interest in mathematic logic higher than I was a student. I have only ...
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### Axiomatic Foundations

I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if ...
1k views

### Mathematicians talking about their identity as a person and as a mathematician? [closed]

I was wondering if any of you know of any books, articles, interviews, youtube videos, ... (etc) where a mathematician talks about his or her identity as a person and as a mathematician? Thank you for ...
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### Trustworthiness of foundational systems

Naively, we might think that if a foundations of mathematics is consistent, then its fair game. Then we learn a bit more, and we realize that even if a foundations of mathematics is consistent, it may ...
988 views

### Don't Gödel's completeness and incompleteness theorems contradict each other? [duplicate]

Gödel's completeness theorem: Given a set of axioms, if we cannot derive a contradiction, then the system of axioms must be consistent. Gödel's incompleteness theorem:'Given any consistent, ...
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### Why is the well ordering principle counter-intuitive?

I read here that while 'The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians'. I don't understand why ...
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### Proofs from the “Ugly Book”

There is a famous saying in mathematics from Paul Erdős: "You don't have to believe in God, but you should believe in The Book." "The Book" is an imaginary book in which God had written down the best ...
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### Constructivism implied or not

Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers? The question was: Let $C$ be a set ...
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### Philosophy of a Mathematician.

Introduction I don't study Mathematics at university, and probably I don't have any chances to have a little understanding of what mathematics in all its aspects. But I love to find structures and ...