Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics ...

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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 ...
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1answer
840 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 ...
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1answer
373 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 ...
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1answer
143 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 ...
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7answers
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 ...
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7answers
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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 ...
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2answers
568 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 ...
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3answers
610 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 ...
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1answer
182 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 $\alpha$ ...
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1answer
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 ...
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4answers
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 ...
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0answers
465 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, ...
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1answer
231 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 ...
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5answers
7k 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 ...
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0answers
104 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 ...
58
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38answers
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 ...
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3answers
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 ...
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1answer
376 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 ...
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0answers
107 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
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2answers
15k 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 ...
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1answer
208 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 ...
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2answers
301 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 ...
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1answer
564 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 ...
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2answers
992 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 $\neg CH$ ...
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21answers
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 ...
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0answers
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 ...
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4answers
7k 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 ...
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5answers
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 ...
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3answers
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 ...
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13answers
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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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5answers
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 ...
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1answer
236 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 ...
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1answer
272 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: ...
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2answers
2k views

Should there be a true model of set theory?

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 ...
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1answer
329 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 ...
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13answers
11k views

Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
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2answers
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 ...
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8answers
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 ...
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9answers
6k views

Is there any formal foundation to ultrafinitism?

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 ...
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1answer
248 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 ...
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2answers
681 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 ...
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2answers
958 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 ...
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7answers
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Extensional theorems mostly used intensionally

Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts: $$ \int_a^b f(x)g'(x)ds = ...
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9answers
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Examples in mirror symmetry that can be understood.

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider ...
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0answers
221 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 ...
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6answers
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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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14answers
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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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10answers
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Is PA consistent? do we know it?

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 ...
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17answers
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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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3answers
428 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 ...