**61**

votes

**39**answers

12k views

### nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...

**28**

votes

**2**answers

3k views

### Why is this new result such a big deal?

This popular article reports a recent result in reverse mathematics, showing that a certain theorem in Ramsey theory is provable from RCA$_0$, the base theory in SOSOA. Then there are a bunch of ...

**2**

votes

**2**answers

348 views

### What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \mid x \notin x\}$ in Frege's Grundgesetze?

In the paper,
Aldo Antonelli and Robert May, Frege's new science, Notre Dame J. Form. Log. 41 (2000), no. 3, 242–-270, MR 1943495.
the authors give the following quote of Frege, from his paper ...

**20**

votes

**2**answers

2k 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 ...

**22**

votes

**3**answers

1k views

### Mathematicians with Aphantasia (Inability to Visualize Things in One's Mind)

Are there any mathematicians with aphantasia? If so, could they please elaborate upon what their experience with mathematics is like?
I realize that this question probably falls outside of the scope ...

**42**

votes

**15**answers

6k views

### Abstract Thought vs Calculation

Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming ...

**49**

votes

**9**answers

6k views

### When have we lost a body of mathematics because errors were found?

The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, ...

**1**

vote

**1**answer

203 views

### Have some works by Émile Borel ever been translated from French to English or another foreign language?

I plan to submit a couple of questions around Émile Borel's works in probability theory to MO.
In this scope, I'd like to know if the following works have ever been translated from French to English ...

**4**

votes

**1**answer

197 views

### Identity types: What makes Intuitionistic Type Theory *intuitionistic*?

In the opening passage of Martin-Löf's (1975) he famously says that
"the theory of types with which we shall be concerned is intended to be a full scale system for formalizing intuitionistic ...

**14**

votes

**1**answer

476 views

### Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?

Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...

**9**

votes

**0**answers

673 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" ...

**8**

votes

**2**answers

756 views

### Is platonism regarding arithmetic consistent with the multiverse view in set theory?

A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a ...

**6**

votes

**2**answers

1k views

### Using the multiverse approach to decide the law of the exluded middle?

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...

**17**

votes

**2**answers

2k views

### Universe view vs. Multiverse view of Set Theory

Here I refer to Hamkins' slides:
http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf
particularly, to the "Universe view simulated inside Multiverse", p. 22.
My question is: is it very unsound ...

**10**

votes

**3**answers

561 views

### The universe of sets, existential quantification in set theory

Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context.
In ZF one can prove $\not\exists x (\forall y (y\in x)).$ ...

**2**

votes

**1**answer

504 views

### Can Turing machines clarify mathematical, philosophical, and physical existence?

From Harvey Friedman's manuscript on "Order Invariant Relations and Incompleteness":
DEFINITION 4.4. A $\Pi_1^0$ sentence is a sentence asserting that some given Turing machine never halts at the ...

**23**

votes

**4**answers

2k views

### In what ways did Leibniz's philosophy foresee modern mathematics?

Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...

**6**

votes

**1**answer

430 views

### Are there 'finitistic' nonrecursive functions (assuming Church's Thesis is false)?

[Note: In what follows, I will be using the same type of argument Laszlo Kalmar did in his paper "An Argument Against the Plausibility of Church's Thesis" found in Constructivity in Mathematics, ...

**27**

votes

**8**answers

2k views

### Interpretation of the Second Incompleteness Theorem

For simplicity, let me pick a particular instance of G\"odel's Second Incompleteness
Theorem:
ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does ...

**28**

votes

**7**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 ...

**1**

vote

**0**answers

206 views

### The theory of frames and locales as elementary topology [closed]

In What is elementary geometry? (pdf) Alfred Tarski defined elementary geometry to be
that part of Euclidean geometry which can be formulated and established without the help of any ...

**1**

vote

**1**answer

172 views

### In what sense is the “descending chain principle” for ordinals less than $\epsilon_0$ 'infinitary?

In the introduction to his paper "Assignment of Ordinals to Terms for Primitive Recursive Functionals of Finite Type", W.A. Howard writes:
Gentzen...showed that the consistency of first order ...

**4**

votes

**0**answers

301 views

### About the “semi-classical” view of Prof. Weaver and Prof. Feferman [closed]

In the thread "Is platonism regarding arithmetic consistent with the multiverse view in set theory?", Prof. Hamkins writes:
The view you are suggesting is something close to what is held by ...

**2**

votes

**1**answer

1k views

### What is the modern consensus on the difficulty of infinitesimals?

At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...

**1**

vote

**2**answers

364 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 ...

**7**

votes

**1**answer

436 views

### Taller models of ZFC

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.
Using forcing techniques, at least the ones I know of, one starts from a ...

**1**

vote

**1**answer

159 views

### What restriction(s) of Goedel's primitive recursive functionals is (are) necessary and sufficient to prove the consistency of $PRA$

It is well known that one can use Goedel's primitive recursive functionals of finite type to prove the consistency of $PA$ (Peano Arithmetic). As such, one can certainly use them to prove the ...

**44**

votes

**7**answers

5k views

### Why hasn't mereology succeeded as an alternative to set theory?

I have recently run into this wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set ...

**28**

votes

**7**answers

3k views

### Can a problem be simultaneously polynomial time and undecidable?

The Robertson-Seymour theorem on graph minors leads to some interesting conundrums.
The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...

**5**

votes

**3**answers

689 views

### What does the axiom of replacement mean and why should I believe it?

Here Professor Blass describes the following cumulative hierarchy of sets:
Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of ...

**3**

votes

**1**answer

193 views

### A question regarding the consistency of Nelson's Predicative Arithmetic

Following Dan Willard (from his paper "Self-Verifying Axiom Systems, the Incompleteness Theorem, and Related Reflection Systems", found on his homepage, pdf here):
"Define an axiom system $\alpha$ ...

**13**

votes

**0**answers

497 views

### Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...

**1**

vote

**1**answer

266 views

### Does mathematical induction presuppose the existence of a completed infinity?

Consider the following statement by Edward Nelson--this from the "Outline" of his 'proof' of the inconsistency of $PA$ (which Terry Tao found to contain an error):
"The induction axiom schema of ...

**32**

votes

**5**answers

8k views

### Categorical foundations without set theory

Can there be a foundations of mathematics using only category theory, i.e. no set theory? More precisely, the definition of a category is a class/set of objects and a class/set of arrows, satisfying ...

**2**

votes

**0**answers

111 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 ...

**53**

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 ...

**9**

votes

**1**answer

394 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 ...

**2**

votes

**0**answers

108 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 ...

**54**

votes

**2**answers

16k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...

**0**

votes

**1**answer

228 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 ...

**0**

votes

**2**answers

311 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 ...

**7**

votes

**1**answer

583 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 ...

**71**

votes

**21**answers

13k 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 ...

**2**

votes

**0**answers

73 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 ...

**58**

votes

**4**answers

8k views

### Nelson's program to show inconsistency of ZF

At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...

**10**

votes

**3**answers

2k views

### Math History Question about the exponential function

While tutoring a student recently, I have come across the situation of explain logarithms by first introducing functions of the form $$f(x)= a^x$$ where $a \ge 0,x\in \mathbb{R}$. My student then ...

**19**

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 ...

**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 ...

**3**

votes

**1**answer

262 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 ...

**4**

votes

**1**answer

278 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: ...