Questions involving philosophy of mathematics

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3
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2answers
190 views

What is the meaning/purpose of finding the “foundations of mathematics”?

I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I ...
0
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1answer
56 views

particular property and completeness?

I was puzzeling with the almost standard definition of completeness: In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula ...
2
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3answers
327 views

How Do You Know If Mathematical Definition Matches Up With Reality?

This is probably one of the biggest question I have when learning some mathematics. I always wonder if I have a concept in my head lets say continuity. Lets I want this concept to be able to ...
1
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1answer
59 views

Formalized philosophy

I once recall a conversation with a friend who told me that his friend was taking a philosophy course where the ideas and concepts were formalized and done very rigorously. This really intrigues me. I ...
0
votes
1answer
74 views

Zero vs Infinity relation type

I'm not sure it should be asked here or in philosophy. Bertrand Russell in his book "Introduction to Mathematical Philosophy" in chapter 7 when discussing rational numbers on page 66 says: "It will ...
7
votes
6answers
411 views

Interviews of famous modern mathematicians

I was wondering, are there any good collections of interviews of famous modern mathematicians? It can be text interviews, or audio or video recordings. I am not sure what exactly I mean by "modern". ...
2
votes
0answers
60 views

Why are divergent Fourier series all so 'HARD'?

I'm not sure if this question is appropriate or even making sense, but I still feel curious: why are every example of divergent Fourier series SO COMPLICATED? It usually takes pages to construct and ...
26
votes
8answers
2k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
11
votes
3answers
666 views

What is more important in Mathematics, Theorems or its Proofs?

Felix Klein once said, Mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof. Till now I thought the opposite. I thought that ...
5
votes
4answers
393 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
0
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0answers
42 views

The Major Weaknesses in Ramified Type Theory

I am reviewing a paper on the major weaknesses of Ramified Type Theory in predicative second-order arithmetic. These four are listed as "weaknesses." But, I have my doubts. It seems at least that 3) ...
3
votes
2answers
166 views

Is it Theoretically Impossible to Demonstrate that Set Theories Are Consistent?

I have to present on the main realist and non-realist arguments for/against set theory. According to one of my sources, it remains a matter of debate as to whether any of the set theories' (ZF, NF, ...
36
votes
14answers
10k views

How big is infinity?

This might be more philosophy than math, but it’s been bothering me for a while. Question: If there’s an infinite amount of real numbers between $ 0 $ and $ 1 $, shouldn’t there be twice the ...
7
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2answers
264 views

Set theoretic realism

What are the main contemporary arguments for and against realism about set theory?
0
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1answer
32 views

Evolution of Relations

In Frege, one finds relations treated as predicates in complex terms. However, modern set theory appears to treat them as two-place relation. Is this correct? If so, when did this shift occur and to ...
1
vote
1answer
100 views

A question regarding Worldly Cardinals and L

For some $L_\kappa$ in the constructible hierarchy, does there exist a $\kappa$ such that $\kappa$ is a worldly cardinal and that $L_\kappa$ contains all of the constructible reals? The motivation ...
1
vote
2answers
104 views

Abstract Objects in Logic

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, ...
2
votes
1answer
100 views

Does math have to be learned linearly?

I am asking because often times one doesn't know where to start in math. "Just learn what you need" is very vague and unspecific ... for example, assume I'm a beginner at Algebra and was considering ...
1
vote
3answers
111 views

Why do we formalize conceptions?

Why do we always try to formalize conceptions? Let's take the naive conception of sets, why do we try to write down a list of axioms? what do we earn in doing so? I'm looking especially for ...
3
votes
1answer
427 views

Do only certain people exceed at math well? [closed]

It's obvious if you look around that math has always been one of the toughest subjects in all areas, from federal-traditional public schools to simply people learning it as an autodidact, hobby, or as ...
2
votes
1answer
70 views

Difference between impredicative and predicative version of separation axiom

What is the difference between an impredicative and a predicative version of the separation axiom in ZFC: $$\forall x \exists y \forall z ( z\in y \leftrightarrow (z \in x \wedge \phi (z)) $$ What ...
2
votes
2answers
100 views

Are axioms chosen with the goal of “making things work” instead of some deep philosophies?

Are axioms chosen with the goal of "making things work" instead of some deep philosophies? If everything should be deducible, that is, provable from something else, then in this chain of deduction ...
21
votes
10answers
2k views

How can Zeno's dichotomy paradox be disproved using mathematics?

A brief description of the paradox taken from Wikipedia: Suppose Sam wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must ...
2
votes
2answers
92 views

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works.

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works. I think Bertrand Russell was a special mind and I set a goal for myself to study all his ...
3
votes
3answers
63 views

Logical 0, binary 0, decimal 0: are they the same?

Logical 0, binary 0, decimal 0: are they all the same in mathematics? A programming language might treat them differently, but is 0 just 0? No matter whether it is logical, binary, decimal, ...
4
votes
3answers
133 views

Disturbing the foundations of mathematics

I was curious of knowing if it is possible that an event "x" could disturb so greatly mathematics that we could be casting doubts on all the achieved results from the very beginning. I'm not sure if ...
2
votes
1answer
160 views

Why should we accept the existence of subsets $A$ such that neither $A$ nor $A^c$ are recursively ennumerable? And how can we persuade others?

Encode every pair $(t,x)$ (where $t$ is a Turing machine and $x$ is an input string) as a distinct natural number. Then the halting subset $H$ fails to be recursive. $$H := \{(t,x) \in \mathbb{N} ...
44
votes
16answers
4k views

Is a proof still valid if only the writer understands it?

Say that there is some conjecture that someone has just proved. Let's assume that this proof is correct--that it is based on deductive reasoning and reaches the desired conclusion. However, if ...
1
vote
1answer
78 views

When is it useful to reduce mathematical objects to foundational levels and when it is not?

When is it useful to reduce mathematical objects to foundational levels and when it is not? Let's say you work in the field of computer vision, or else. How can you claim your method is optimal if ...
1
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0answers
50 views

Trying to understand Hintikka's logic of Knowledge and belief

I try to understand Hintikka's logic of knowledge and belief but am a bit stumped by it. I study " Knowledge and belief , an introduction to the Logic of the two Notions", (Kings College ...
4
votes
3answers
163 views

Are there other approaches for the foundations of mathematics, other than logic and set theory?

Are there other approaches for the foundations of mathematics, other than logic and set theory? And why does set theory begin talking about objects and groups of objects. Is it proven somewhere that ...
0
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1answer
82 views

If a set is a group of objects, then what is an object? [closed]

If a set is a group of objects, then what is an object? My best try at this is the following: An object is anything that we can discuss or think about, separately from everything else. It is not ...
2
votes
1answer
63 views

Question about the univalence axiom versus skeleta.

Here, Dan Licata writes: [Univalence] can be used to build algebraic structures in such a way that isomorphic structures are equal (e.g. equality of groups is group isomorphism). He writes ...
1
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1answer
51 views

Constructivist Interpretation of a Function

Lets suppose I have an exponential function $a^{x}$, and I desire to show that for any number $n$ in $(0, \infty)$, it is possible to find a value of $x_0$ such that $a^{x_0} = n$. The simplest proof ...
1
vote
2answers
144 views

Consistency of ZFC and the key assumption [closed]

I recently read this answer to a MathOverflow question that got me thinking. Very roughly, here's what the author of that answer says: Gödel's second incompleteness theorem implies that if there ...
2
votes
1answer
110 views

Do circles exist

So I was wondering about circles today and if they really do exsist. If you graph a circle in function mode, your equation looks like$$y=\sqrt{1-x^2}$$ Now for simple purposes lets take a portion of ...
7
votes
2answers
235 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
2
votes
1answer
54 views

Variable theory

I wanted to know if there's any alternative variable theory to Russell's used in his Principia. I mean, the modern definition of "variable" and "constant" still follows his works? I try to search on ...
4
votes
3answers
192 views

Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?

It is frequently stated (in textbooks, on Wikipedia) that the "Law of large numbers" in mathematical probability theory is a statement about relative frequencies of occurrence of an event in a finite ...
3
votes
6answers
212 views

Logical issues with the weak law of large numbers and its interpretation

In several probability textbooks I have found what amounts to the following argument: Let A be an event in some probabilistic experiment. Let p=P(A) be the probability of this event occurring in ...
1
vote
2answers
80 views

How an axiomatic system is made?

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call ...
5
votes
4answers
150 views

“The enrapturing discoveries of our field systematically conceal (…) the analogical train of thought that is the authentic life of mathematics…”

In the preface of the book Discrete Thoughts, Gian-Carlo Rota writes: Sometime, in a future that is knocking at our door, we shall have to retrain ourselves or our children to properly tell the ...
3
votes
2answers
91 views

How does undecidability of 'theoremhood' imply that human ingenuity is necessary in mathematics?

In Robert Stoll's "Set Theory and Logic", there is the following passage on effectiveness of theorems (p. 375) : Mathematical logicians have shown that for many interesting axiomatic theories ...
3
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1answer
164 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
2
votes
0answers
70 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
3
votes
1answer
299 views

First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second ...
1
vote
1answer
60 views

What exactly does $\vdash_T G_T \leftrightarrow \lnot \exists y$ Prf$(\ulcorner G_T \urcorner, y)$ mean?

To me this translates to: $G_T$ is provable in $T$ if and only if there doesn't exist a $y$ such that $y$ is a witness to the provability of $\ulcorner G_T \urcorner$. But I'm not entirely sure what ...
6
votes
3answers
242 views

Is math independent of our sensory experience? [closed]

I've been asking myself this and other questions in the field of philosophy of mathematics. Could we, if we were isolated from any kind of sensory experience, be able to learn mathematics? Also, what ...
1
vote
4answers
150 views

Where does the importance of math come from? [closed]

It is a somewhat philosophical question. I personally believe that the importance of math is due to its usefulness and lots of applications. Mathematics is used in everywhere nowadays; as Ian Stewart ...
1
vote
1answer
59 views

Sheafs appearing in philosophy?

I apologize in advance if I make mistakes in the following construction. I have very recently been introduced to the concept of a sheaf. I am currently a mathematics major and philosophy minor and ...