**2**

votes

**0**answers

30 views

### Some questions regarding an alteration of Grzegorczyk's theory of concatenation, $\operatorname{TC}$

Consider Grzegorczyk's concatenation theory $\operatorname{TC}$, a "weak theory of words over the two letter alphabet $\Sigma=\{a,b\}$" (this from Grzegorczyk and Zdanowski's paper Undecidability and ...

**2**

votes

**0**answers

146 views

### Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?

Was "arithmetical translation" (that is, coding in the Goedel sense) ever a part of Hilbert's Program? I ask this question for several reasons:
i) it gives the numerals |, ||, |||,.... an ersatz ...

**9**

votes

**1**answer

358 views

### Examples of abstractions that did *not* turn out to be useful [closed]

I’ve read (but cannot find any reference now) that new abstract mathematical concepts like set theory and – not too long ago – category theory were in their time often considered too abstract to be ...

**52**

votes

**3**answers

4k views

### What was Hilbert's view of Gödel's Incompleteness Theorems?

According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
...the end goal [is] to establish as ...

**2**

votes

**0**answers

100 views

### A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability

Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...

**2**

votes

**0**answers

72 views

### Which self-reference restrictions can be weakened in probabilstic logic?

This work suggests that there is some generalization of Truth in terms of probability, which can be definable within the logic itself.
Is where any other thorems on self-reference restrictions, which ...

**22**

votes

**5**answers

2k views

### Why should we believe in the axiom of regularity?

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of ...

**0**

votes

**2**answers

292 views

### A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals

A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there ...

**3**

votes

**1**answer

222 views

### Information theory from negative probability

Szekely provides a convincing argument of negative probability here:
http://www.wilmott.com/pdfs/100609_gjs.pdf
What does a reformulation of classical information theory built from negative ...

**0**

votes

**1**answer

203 views

### A question regarding extendible cardinals and a result of M. Magidor

The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971):
"Definition: Logic is called ...

**4**

votes

**1**answer

268 views

### Plausibility argument for a measurable cardinal

The following question is not mathematically precise but perhaps of some philosophical interest.
A typical plausibility argument for assuming the existence of inaccessible cardinals goes as follows: ...

**1**

vote

**1**answer

317 views

### Is second-order ZFC categorical with regard to its proper class models

Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...

**29**

votes

**8**answers

5k views

### Uninteresting questions with interesting answers [closed]

What are best examples of questions in mathematics that are not interesting until one knows the answers, whose answers themselves are what is interesting?
The thing that prompts me to post this is ...

**17**

votes

**2**answers

1k views

### Philosophical arguments in defense (or against) large cardinals

The question is essentially what is asked in the title. I split it into two parts
(A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense ...

**7**

votes

**1**answer

240 views

### Is $ACA_0$ + `True Arithmetic exists' interpretable in $ACA$?

Maybe someone here can help me with a question concerning second-order arithmetic. Consider the system $ACA_T := ACA_0 + \exists X \forall x (x \in X \leftrightarrow T(x))$, where $T(x)$ is a ...

**14**

votes

**2**answers

938 views

### nonstandard models and mathematical theorems

Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard ...

**10**

votes

**2**answers

603 views

### Is there a compendium of the consistency strength between the most important formal theories?

Preliminar Notions:
A formal system is a tuple $(\Sigma,G,A,R)$ where $\Sigma$ is an alphabet (set of symbols), $G$ is a formal grammar on $\Sigma$ that generates a formal language $L$ (set of well ...

**2**

votes

**0**answers

200 views

### Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given
by ...

**6**

votes

**3**answers

416 views

### Extensionality in HoTT versus extensionality in internal language of a category

What's the extension of judgmental identity in HoTT (homotopy type theory)?
The Martin-Löf intensional dependent type theory with identity types is called (definitionally) extensional if the ...

**70**

votes

**21**answers

12k views

### Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...

**8**

votes

**2**answers

1k views

### The impact of large cardinals in mathematics [closed]

What are the main applications of large cardinals in ordinary mathematics, and what is the philosophy behind using them. In particular:
Question 1. What is the philosophy behind accepting large ...

**12**

votes

**1**answer

711 views

### Time in Girard's Geometry of Interaction

Jean-Yves Girard writes at the end of his paper
"Towards a Geometry of Interaction", page 105, that we have three intuitions about the nature of time:
time is logic modulo the order of rules,
time ...

**-4**

votes

**1**answer

249 views

### Universal quantifier in Russell's Theory of descriptions - Who is the UNIVERSE? [closed]

To moderators: Please don't delete or migrate this thread: It's by no means PURE PHILOSOPHICAL, but mostly a mathematical logic question!
In Russell's 1905 paper "on denoting" in which he introduces ...

**6**

votes

**2**answers

563 views

### Can we define an “empirically generic” real number?

Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...

**5**

votes

**2**answers

1k views

### Meta$^{n{-}th}$ mathematics [duplicate]

Metamathematics has a reasonably clear connotation,
enough to have a Wikipedia page,
with Gödel, Tarski, and Turing playing leading roles;
Kleene's book (Introduction to Metamathematics (Amazon ...

**19**

votes

**5**answers

2k views

### What's special about the Simplex category?

I have been wondering lately what makes simplicial sets 'tick'.
Edited
The category $\Delta$can be viewed as the category of standard $n$-simplices and order preserving simplicial maps. The goal of ...

**0**

votes

**0**answers

104 views

### Geometric interpretation of table with permutations and inversions

Let $T(n,k)$ is the number of permutations of numbers $1, ..., n$ and each of the permutations has $k$ inversions. We can consider a table for $T(n,k)$ for some $n$ and $k$. For eg.
$n=1,...,6$, ...

**13**

votes

**3**answers

525 views

### Formal/rigorous treatment of (im)predicativity/predicativism

There are several places on the web where one may find quite intuitively understandable accounts of (im)predicativity; here on MO I found two questions with very good detailed answers (Predicative ...

**7**

votes

**1**answer

556 views

### Last Status of Feferman's Conjecture on Indefinite Value of Continuum

The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...

**15**

votes

**6**answers

2k views

### What “forces” us to accept large cardinal axioms?

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).
Their non-existence is consistent with axioms of usual mathematics.
It is provable that some of ...

**10**

votes

**5**answers

2k views

### Interesting meta-meta-mathematical theorems?

The Goedel incompleteness theorems can be considered meta-mathematical theorems, as they are "written" in a meta-theory and "talk" about properties of a class of formal theories.
The following may be ...

**1**

vote

**1**answer

664 views

### What are trivial objects, in general?

Trivial objects show up in most every branch of mathematics, and we all know lots of examples: the trivial group, ring, vector space, module over a ring, graph, knot, homomorphism from one object to ...

**2**

votes

**3**answers

1k views

### Assessing effectiveness of (epsilon, delta) definitions [closed]

There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...

**3**

votes

**2**answers

299 views

### Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book
Robinson, A.; Laurmann, J. A. Wing ...

**2**

votes

**0**answers

126 views

### Is there an universal (dis)similarity measure between two structures?

I'm always wondering is there an universal (dis)similarity measure
between two structures (let's say between two undirected simple
graphs)? I mean, not "the measure with universal parameter that we
...

**4**

votes

**1**answer

367 views

### Fundamental Problems in Mathematics that, without Computer Sciences, would not be resolved? [closed]

Could you please give examples of fundamental questions in mathematics (let us say, pure mathematics) which were resolved fundamentally by the use of computers? More precisely, are there examples that ...

**3**

votes

**2**answers

522 views

### Are descriptive and ontological notions of equality equal? [closed]

Let $a$ and $b$ are two "objects". What is the meaning of $a=b$? This is one of the deepest problems of philosophy and logic because one needs a complete information about ...

**8**

votes

**0**answers

550 views

### Has anyone pursued Frege's idea of numbers as second-order concepts?

Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" ...

**18**

votes

**13**answers

2k views

### An example of a proof that is explanatory but not beautiful? (or vice versa)

This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs ...

**10**

votes

**2**answers

573 views

### Are simplicial sets the intended model of HoTT?

While thinking about Jason Rute's question, I wondered if there was an intended model for HoTT. The main candidate for the intended model are simplicial sets, where Vladimir Voevodsky first observed ...

**5**

votes

**2**answers

500 views

### What is the impact on Godels theorem of Paraconsistency?

Russells paradox forced a restriction of the natural abstraction principle (that every predicate determines a set) so that Set Theory could be consistent. The standard one being ZF.
However ...

**1**

vote

**0**answers

108 views

### A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)

In their paper "The Theory of Sets of Ordinals" (arXiv), Koepke and Koerwien propose a theory SO axiomatizing the class of sets of ordinals in a model of ZFC and show that SO and ZFC are ...

**1**

vote

**2**answers

570 views

### Ontological status of some “sets” in ZFC [closed]

Let $\phi$ be an undecidable statement of ZFC set theory, for example let's take continuum hypothesis.
What is the ontological status of the "set" $X=\bigl\{x\in\{1,2\}:x=1\text{ or }(x=2\text{ and ...

**2**

votes

**2**answers

1k views

### In What Sense is Set Theory a 'Foundation' for Mathematics? [closed]

In what sense is set theory a foundation for mathematics? To my mind (for what that is worth), there are at least three (somewhat) distinct senses in which set 'theory' (I put "theory" in scare ...

**46**

votes

**7**answers

2k views

### How closed-form conjectures are made?

Recently I posted a conjecture at Math.SE:
$$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J_\mu(x)$ and $Y_\mu(x)$ ...

**10**

votes

**6**answers

1k views

### Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with intuitionistic logic. It is ...

**17**

votes

**2**answers

916 views

### Age of Stochasticity?

One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here.
The question is this:
Today ...

**44**

votes

**17**answers

9k views

### Is rigour just a ritual that most mathematicians wish to get rid of if they could?

"No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was one of the panelists ...

**9**

votes

**3**answers

911 views

### Is there an observer dependent mathematics? [closed]

Is there any field of mathematics that deals with the role of the observer? E.g., some formulation in which a set is changed, in some unspecified way, when it is observed? Or maybe some philosophy of ...

**14**

votes

**1**answer

1k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...