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1answer
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Analogue of prime numbers in addition? [closed]

What is the analogue of prime numbers in addition?
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1answer
235 views

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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0answers
116 views

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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3answers
148 views

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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1answer
143 views

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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0answers
548 views

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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1answer
300 views

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 ...
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7answers
851 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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1answer
572 views

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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1answer
72 views

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 ...
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3answers
140 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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250 views

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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7answers
734 views

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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0answers
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Stylish Academic Writing [duplicate]

I don't know that this question belongs here, but I'd like to know of any references out there anyone here might recommend for writers of mathematical ideas, be it a book, an article, a dissertation, ...
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2answers
56 views

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 ...
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1answer
642 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, ...
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1answer
192 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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2answers
420 views

Books about modal logic?

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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5answers
974 views

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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6answers
368 views

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 ...
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4answers
2k views

Is mathematical history written by the victors?

The question is the title of a recent piece in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is ...
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9answers
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 ...
4
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1answer
108 views

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 ...
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2answers
615 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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3answers
837 views

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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2answers
609 views

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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3answers
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Yablo's paradox? a paradox without self-reference [closed]

Yablo's paradox arises from considering the following infinite set of sentences: $$(S_1): \mbox{for all }k > 1, S_k\mbox{ is false} \\ (S_2): \mbox{for all }k > 2, S_k\mbox{ is false} \\ ...
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What leads us to believe that 2+2 is equal to 4? [closed]

My professor of Epistemological Basis of Modern Science discipline was questioning about what we consider knowledge and what makes us believe or not in it's reliability. To test us, he asked us to ...
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1answer
121 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 ...
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1answer
277 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
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3answers
182 views

There is concept of finite sets that can have only one “interpretation”?

In our mind we have a naive idea of what a set is, and in nature we can only observe something that behave like a finite set, ZFC (or set theories in general) tries to catch these properties in ...
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2answers
315 views

Why demonstrations are important in mathematics? [closed]

Good evening, I'm studying math and would like to know how important are mathematical proofs in the world and particularly in a school of mathematics Thanks for your help
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2answers
144 views

The consistency of PA is falsifiable. Can the same be said of its soundness?

The statement that 'PA is consistent' is absolutely falsifiable (a term I just made up), in the sense that if it is false, then we can demonstrate its falseness independently of any metatheory (just ...
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3answers
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What would be the immediate implications of a formula for prime numbers?

What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
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1answer
356 views

Binary vs. Ternary Goldbach Conjecture

Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current ...
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9answers
4k views

How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
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3answers
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Axiom of Choice-esque argument to show that a proof of a statement exists without actually giving a proof

What if the set of all well-formed statements in ZFC formed a kind of pseudo-category where a morphism f between objects A, B represented a formal proof that A implied B? What if that category could ...
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4answers
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Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
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1answer
748 views

What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions?

I got the number $$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$ in the ...
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2answers
196 views

Truth of Fundamental Theorem of Arithmetic beyond some large number

Let $n$ be a ridiculously large number, e.g., $$\displaystyle23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23}}}}}}}}}}}}+5$$ which cannot be explicitly written down provided the size of the ...
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2answers
822 views

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 ...
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25answers
14k 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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2answers
654 views

Reinterpreting improper integrals that require Cauchy principal value to be defined

This question concerns the Cauchy principal value. Consider the improper integral $$\int_{-∞}^{∞}\frac{1+x}{1+x^2}dx$$ which is divergent, and then its Cauchy principal value $$\lim_{u \to ∞} ...
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5answers
914 views

What does it mean for a set to exist?

Is there a precise meaning of the word 'exist', what does it mean for a set to exist? And what does it mean for a set to 'not exist' ? And what is a set, what is the precise definition of a set?
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2answers
219 views

Gray's “Plato's Ghost” - a curious mistake

I am currently reading Jeremy Gray's "Plato's Ghost", and I run into the following passage (Chapter 5, page 332). The point is, it seems to me that it contains two very elementary mistakes that feel ...
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1answer
181 views

Does ZFC have an intended interpretation?

I know that PA has an intended interpretation, namely $\mathbb{N}$, and the usual axioms of the real line have an intended interpretation, namely $\mathbb{R}$. Does ZFC have an intended ...
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278 views

Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?

I've heard people make the argument that: $\mathsf{ZFC}$ suffices as a foundations of mathematics because almost all theorems in the mathematics literature can be proven using $\mathsf{ZFC}$, so ...
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6answers
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What is this physicist saying?

I do not want to poison this forum with politics. But I want to understand, precisely, what is meant by the bolded statement. It is made by a physicist who used to work at Harvard regarding the ...