**250 mathematical-philosophy questions.**

This question follows up on an issue arising in Peter LeFanu
Lumsdaine's nice question: Does foundation/regularity have any
categorical/structural consequences, in
ZF?
Let me mention first that my ...

Does the fact that, assuming the consistency of $ZFC$, no proof that the consistency of "$ZFC$ implies the consistency of '$ZFC$ + There exists a weakly inaccessible cardinal'" can be formulated in $...

The search for a neat Theory of Everything (ToE) which unifies the entire set of fundamental forces of the universe (as well as the rules which govern dark energy, dark matter and anti-matter realms) ...

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 foundation/...

Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was the use of more abstract notions ...

Starting to write an introduction to the philosophy of mathematics I find tons of positions that are of historical interest. My question to the research community in mathematics is which positions are ...

Ten years ago, when I studied in 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 ...

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs (...

This question is in connection with the question that I've asked at:
Where do models of false theories exist?
The answer to that question was that any consistent theory can have its primitives be re-...

I have some difficulties in understanding the [uni]verse platonic view. How are we to understand the existence of a model of a false theory? what is the relationship of this model to the platonic ...

The topic of this post was shifted to
https://philosophy.stackexchange.com/questions/49504/is-it-natural-to-hold-that-big-sets-and-proper-classes-exist
Since it was deemed to be a philosophical ...

It was an ambitious project of Vladimir Voevodsky's to provide new foundations for mathematics with univalent foundations (UF) to eventually replace set theory (ST).
Part of what makes ST so ...

Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to wikipedia, it has been ...

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...

The purpose of my question is trying to understand whether, in some cases, we can achieve greater certainty of reasoning (say when dealing with statements about natural numbers, integers or rational ...

It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:
What are some ...

The Singular Cardinal Hypothesis (SCH) is the statement that $\kappa^{cf(\kappa)} = \kappa^+ \cdot 2^{cf(\kappa)}$ for every singular cardinal $\kappa$ (or various equivalent statements).
It is ...

A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent ...

"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 ...

(Note: the 'Second-order $ZFC$' ($ZFC^2$) I am talking about is the theory [in the second order language of set theory consisting of a single non-logical symbol $\in$ ] consisting of the axioms ...

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 ...

This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...

Return to Frege's question, What justifies arithmetic? And consider the ur-proposition that counting a finite set always produces the same number, and ask whether this has a logical justification, ...

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 ...

I seem to remember reading once a story that some mathematician had written to justify the use of categories, or isomorphisms or equivalences, or something like that. The story goes something like ...

According to constructivism, "it is necessary to find (or "construct") a mathematical object to prove that it exists". There are several formulas to calculate $\pi$, such as:
...

Edit (15 November 2017): I've just stumbled across this question, which I think is asking essentially the same thing I tried to ask below, but probably worded it more clearly - and got far more ...

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 ...

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 ...

Recently I have "finished" a 13-year on and off research on the history of the mathematical notion of equivalence. At the end of which, I learned that we owe the nowadays rather elementary process of "...

If definitions themselves are informally just maps from words to collections of other words. Then in order for one to define anything, they must inherently already have a notion of a function. I mean ...

One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, ...

It is a matter of mathematical folklore that Gödel "entertained the idea of so called stronger axioms of infinity deciding $CH$...." (this quote from Radek Honzik's paper, "Large cardinals and the ...

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 ...

My first language is not English. How can I improve my mathematical writing. I feel like the only things I can write down are numbers and equations. Is there any good suggestion for improving writing, ...

I am interested in asking the following question:
What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation ...

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 ...

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 ...

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 ...

Here, Noah Schweber writes the following:
Most mathematics is not done in ZFC. Most mathematics, in fact, isn't done axiomatically at all: rather, we simply use propositions which seem "intuitively ...

In
Lawvere, F. W., 1966, “The Category of
Categories as a Foundation for
Mathematics”, Proceedings of the
Conference on Categorical Algebra, La
Jolla, New York: Springer-Verlag,
1–21.
...

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 ...

In Lecture Notes in Mathematics 1488, Lawvere writes the introduction to the Proceedings for a 1990 conference in Como.
In this article, Lawvere, the inventor of Toposes and Algebraic Theories, ...

Where can you find Grothendieck's "Recoltes et Semailles"?
Is it available anywhere?

As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "On the Infinite". It is also known that it is false.
Has there ever been a published ...

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 ...

This question is probably really naive. And, I hope the title doesn't come off as too combative. I think that topoi of $\mathbf{Set}$-valued sheaves provide an excellent motivation for higher-order ...

I have heard the perspective that chromatic homotopy theory is meant to break apart the stable homotopy groups of spheres into manageable pieces, and that this perspective has led to various insights. ...

Consider the topological set theory $GPK^{+}_{\infty}$ +$AC_{WF}$, where $GPK^{+}_{\infty}$ is defined as follows (this from Olivier Esser's papers, "An Interpretation of the Zermelo-Fraenkel Set ...

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 ...

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