The math-philosophy tag has no wiki summary.

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### Different approaches to the multiverse of sets

There are some different approaches to the multiverse of sets, in particular:
1) The approach by Woodin,
2) The approach by Sy Friedman, ...,
3) The approach by Hamkins.
I wonder to know if ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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**1**answer

309 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 ...

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### 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 ...

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### 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" ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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508 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 ...

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### 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 ...

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### 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 $Y_\mu(x)$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### A question regarding Koepke' s Ordinal Computability in HOD

Consider the following theorem of Koepke-Koerwien-Siders:
"A set x of ordinals is ordinal computable [either by ordinal Turing machines or ordinal register machines--my comment] if and only if it is ...

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### Non-constructive proofs vs. efficient algorithms

My question concerns what is meant by "nonconstructive", and whether it has ever been defined in terms of computational complexity.
The wikipedia article on constructive proof begins, "a constructive ...

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### Excellent mathematical explanations

In the Stanford Encyclopedia of Philosophy there is an entry on mathematical explanation.
The basic philosophical question is: What makes a proof explanatory?
Two main "models" of mathematical ...

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### Statements which were given as axioms, which later turned out to be false.

I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness ...

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### A Question Regarding Boolean-valued Models

What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...

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### Sets = structured sets without structure

Motivation
There is presumably no single and widely accepted formal definition of structured sets = sets plus structure based on sets as primitive objects, but several approaches are around. See e.g. ...

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### Has the notion of “space” been reconsidered in 20th century?

The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry"， an English version ...

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### Believing the Conjectures

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...

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### Are the Foundations of Mathematical Logic Shaky? [closed]

The mathematics community at large seems pretty satisfied right now with the common practice of 1. starting with some axioms and 2. deriving theorems from them by employing some logic. All mathematics ...

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### How to tell a paradox from a “paradox”?

Russell's paradox showed that naive set theory leads to a contradiction. This was something that was taken seriously and caused a lot of work.
Now, Banach–Tarski paradox is arises from a result that ...

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### Does there exist a non-trivial Ultrafinitist set theory?

Does there exist a set theory T-which has not yet been proved to be inconsistent-and in which
one can prove the existence of (1) the empty set (2) sets that are singletons and (3) sets which
have ...

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### 2D Problems Which are Easier to Solve in 3D

It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...

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### intensional equality in type theory

I want to know why we add an intensional equality in type theory to definitional equality ?
What is the aim with this intensional equality ?
thanks

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### sine and Archimedes' derivation of the area of the circle

The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College ...

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### What is the status of irrational numbers within finitism/ultrafinitism?

According to constructivism a mathematical object to prove that it exists". There are several formulas to calculate pi, such as:
so I take it ...

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### Proof by `universal receiver'

Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually ...

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### ULTRAINFINITISM, or a step beyond the transfinite

Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE.
$\aleph_0, \aleph_1,\aleph_2\dots$
the lists ...

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### Arguments against Reductio ad Absurdum [closed]

Could Reductio ad Absurdum not be consireded a valid proof method? Are there any compelling arguments against it, or at it's favor?
I feel like I am assuming some metamathematical hypothesis about my ...

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**1**answer

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### comprehension and ideal elements

A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and ...

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### Have we ever lost any mathematics?

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, ...

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**1**answer

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### Mathematical analysis of Lewisian concepts, esp. natural properties

David Lewis was one of the great philosophers of our time. He was a genuine philosopher, his focus was on theoretical metaphysics. And he had something to say about mathematics. His last book - he ...

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### Essential reads in the philosophy of mathematics and set theory

I am graduate student and have a decent understanding of logic and set theory.
Recently I have got interested in the philosophy of mathematics and set theory. I have read a number of papers by ...

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### Which mathematical questions are objectively true or false? [closed]

Which parts of mathematics are objectively true or false like finite arithmetic and which are only true, false or undecidable relative to a particular axiom system such as Euclidean geometry?
...

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### Is Algebraic Geometry really natural? [closed]

Dear All!
I recently had a conversation with one mathematician who reckons that all sorts of combinatorial results are nothing compared to the things done in the algebraic geometry. As I do not have ...

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### History of Logic Development

Where can I find a book which explains the development of modern logic, e.g. Tarski, Frege, Peano, up untill Wittgenstein, Russel?

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### a priori grounds of mathematics [closed]

Hi,
from the title of the question you may guess I'm new to the site, and I am. Even though I've read the FAQ and I know that this isn't the place for such open questions (I believe it is open, but ...

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### Variable-centric logical foundation of calculus

Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For ...

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### 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 ...

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### Is there a finite-dimensional vector space whose dimension cannot be found? [closed]

Is there a finite-dimensional vector space whose dimension cannot be found? Assume, we have somehow constructed a vector space whose dimension is finite, but yet unknown. Is there always an algorithm ...