Tagged Questions

Questions involving philosophy of mathematics

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2
votes
1answer
199 views

Did large cardinals exist before 1963?

I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists ...
5
votes
1answer
229 views

What do ultrafinitists think about Graham's number?

I know ultrafinitists want to require not only that mathematical objects be constructible, but be constructible given finite resources (such as time). So I wonder about something like the famous ...
7
votes
1answer
156 views

Set theoretic implications of constructions in Differential Geometry/ Topology

In subjects like Differential Geometry/ General Topology one often constructs for each $x$ in a space $X$ a set $U_x$ satisfying certain properties. Examples where one does constructions like this: ...
2
votes
1answer
112 views

Is there a probability interpretation that only allows for probabilities in $\left[0,1\right]\cap\Bbb Q$?

Are there probability interpretations that only allow for probabilities that are members of the set $$\left[0,1\right]\cap\Bbb Q?$$ Related, but distinct: Allowed probabilities under frequentism.
4
votes
2answers
89 views

Allowed probabilities under frequentism

Am I right to assume that under the frequentist interpretation of probability,* the set of allowed probabilities isn't $$\left[0,1\right],$$ but rather is ...
3
votes
1answer
142 views

Formality fades away in the air

I'm trying to study by myself mathematics, but I realized that I have only a naive notion of certains building blocks of mathematics; certain parts of the formalism. So I tried to start with logic, ...
18
votes
1answer
410 views

Are there areas of mathematics (current or future) that cannot be formalized in set theory?

I often read that ZFC can formalize "most" of everyday mathematics, but I could never find an example which it cannot. The closest I got is differential geometry (DF), where some article mentions that ...
2
votes
2answers
207 views

Is all mathematics based on the concept that $1+1=2$?

Thought about this recently, and was a bit stuck. Is all mathematics based on the concept that $1+1=2$? For example, if $1+1\ne2$, then all arithmetic won't work, right?
4
votes
2answers
231 views

why does soundness seem to be less important than consistency for the structuralist?

If I am not wrong, many mathematicians (I believe this is not only restricted to structuralists) agree that an inconsistent formal system does not have any model. By model I mean some kind of set ...
0
votes
0answers
107 views

Does it become more likely that ZFC is consistent, the more time we explore it without finding a contradiction?

Intuitively, the more time we spend exploring ZFC without finding a contradiction, the higher the (subjective) probability that ZFC is consistent. Is this intuition sound? If not, why not?
1
vote
1answer
1k views

Is mathematics considered a science [duplicate]

Possible Duplicate: what is the definition of Mathematics ? I would like to know if mathematics is considered a science? I've searched the internet and asked many people for insight to no ...
5
votes
10answers
929 views

what is the definition of Mathematics ?

we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically but the question is, what is mathematics ? how can we ...
3
votes
5answers
485 views

About mathematics and the physical world

Suppose it is proven that in the physical universe all magnitudes are finite: there are no infinitely long magnitudes. there are no infinitely small magnitudes. Then: Would we get a mathematic ...
6
votes
1answer
300 views

Age of Stochasticity?

Today I came across D. Mumford's 1999 article The Dawning of the Age of Stochasticity, which is quite remarkable even after more than a decade. The title already indicates the theme, but I copy the ...
-1
votes
4answers
261 views

The facts about $\varphi$ [closed]

A lot of people believe there is something special about the number $\varphi= \frac {1+ \sqrt5}{2}$. However, I can only think of cultural explanations for looking at each property of $\varphi$ as ...
8
votes
5answers
546 views

Successful approaches to the modelization of ''randomness''

If you pick a number $x$ randomly from $[0,100]$, we would naturally say that the probability of $x>50$ is $1/2$, right? This is because we assumed that randomly meant that the experiment was to ...
2
votes
2answers
323 views

What is the “shape” of numbers in Number Theory?

While reading popular science book Fermat's Last Theorem I was amazed to find out that in number theory interesting things happen even at very large scales. For instance the Graham's number was named ...
1
vote
3answers
483 views

Looking for philosophical subject for my Bachelor Thesis

In may 2013 I have to write a Bachelor Thesis for my bachelor Mathematics. I prefer to choose a subject which involves philosophy. At the same time I have the feeling that my university wants me to ...
8
votes
5answers
2k views

Mathematics, Philosophy and writing.

Do you know of any famous mathematicians who were also philosophers? I have heard of Descartes, Plato and Leibniz. Are there other good examples, especially more modern examples? Also welcome are ...
0
votes
1answer
124 views

Infinite versus unendlich and double-negation

The German term for infinite is unendlich, which transliterates as non-ending, or non-finite. This is just word-play but from a constructive point of view, is the shift from a negative to a positive ...
3
votes
2answers
257 views

Books on the philosophy of mathematics and logic

Here is a list of some books on the philosophy of Mathematics and logic founded in an article about this matter. I would like to buy one or maybe two of these or any other suggested books. I would be ...
3
votes
4answers
300 views

Why $\sqrt{\frac {\sum(x-\mu)^2} {N}}$ instead of $\frac {\sum{\Bigl|x-\mu\Bigr|}} {N}$? [duplicate]

Possible Duplicate: Motivation behind standard deviation? In statistics very often you see something of the sort: $$ \textrm{quantity}=\sqrt{\frac {\sum(x-\mu)^2} {N}} $$ to measure things ...
6
votes
2answers
476 views

Could computers someday discover theorems or find demonstrations?

Cloud computing and quantum computers bring computers to what seems like a limitless calculation power? If one sees all the mathematical operations and theorems as a toolset that a computer can use, ...
9
votes
1answer
179 views

Is there an online Q/A forum for the philosophy of mathematical practice?

Certain philosophers of mathematics are interested in aspects of the philosophy of mathematical practice. Mathematicians, perhaps, would be interested in philosophy that may affect their day to day ...
13
votes
4answers
766 views

How do mathematicians think about the existence of numbers?

Question: How do mathematicians think about the existence of numbers? And how did Newton, Euler, and other famous mathematicians thought about this concept? I know that existence of numbers is a ...
5
votes
4answers
382 views

Logic as subset of mathematics and mathematics as subset of logic

Is logic a subset of mathematics or is mathematics a subset of logic? I have heard the former view, but is there any argument for the latter?
2
votes
2answers
202 views

What does Russell mean when he defines the “Posterity… with respect to the immediate predecessor”?

The the Introduction to Mathematical Philosophy, Russell defines the "posterity" of a given number with respect to the relation "immediate predecessor" as all those terms that belong to every ...
8
votes
4answers
701 views

What would happen if ZFC were found to be inconsistent?

If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that ...
64
votes
4answers
7k views

How do I convince someone that $1+1=2$ may not necessarily be true?

Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical ...
5
votes
2answers
125 views

On the existence of number systems, and the extent to which we can extend them

The more I think about math, the less I realize I know. Learning about complex numbers has called me to re-evaluate how I think of negative numbers, or even natural numbers. I have to say the ...
8
votes
4answers
437 views

What is straight line?

I have found the definition of line in metric space. It is general but has two problems. Considering about $\mathbb R^2$ equipped rectilinear distance, every line by this definition contains a ...
-2
votes
2answers
128 views

Is math the measurement of motion? [closed]

I know that math can be used to measure motion, like where something will end up over a period of time, but is math itself 'in' motion. Take a look at how you use math: You scribble, type, draw ...
5
votes
6answers
2k views

How to interpret material conditional and explain it to freshmen?

After studying mathematics for some time, I am still confused. The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the ...
1
vote
1answer
83 views

A Simpler Characterization of Inductive Definitions?

While reading appendix A of John Harrison's "Handbook of Practical Logic and Automated Reasoning" a somewhat advanced theorem is appealed to as a prerequisite for characterizing when an inductive ...
4
votes
2answers
274 views

What is the role of Topology in mathematics?

What is the role of Topology in Mathematics? Is it like Logic that you need in every areas of math?
2
votes
2answers
271 views

What is the use of such concepts as potential infinity and actual infinity?

I'm aware of such mathematical concepts as and potential infinity and actual infinity. But I do not understand how those concepts are being used. Are there any applications to such concepts? Are there ...
2
votes
3answers
130 views

Polynomials as concrete structures

Motivation The structuralist point of view on mathematical objects has two aspects: On the one side, a mathematical object is seen as a concrete structure of abstract dots, e.g. a graph. On the ...
6
votes
3answers
1k views

are there non-standard models of arithmetic in second order arithmetic?

non-standard models of arithmetic in second order arithmetic? Background: According to Godel's theorem, if we have, in a given consistent system S, a non-provable wff. A, then we can extend the ...
6
votes
4answers
309 views

Is probability objective?

As we know, probability is a measure of events. However, is it an objectively attribute of events, or just an illusion in ones' mind? For example, suppose that there is an empty black box with an ...
48
votes
10answers
12k views

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
13
votes
2answers
560 views

Is First Order Logic (FOL) the only fundamental logic?

I'm far from being an expert in the field of mathematical logic, but I've been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and ...
9
votes
4answers
491 views

What are the reasons for not supporting constructive mathematics

It is obvious that in constructive mathematics, you cannot use the law of excluded middle. What else would be the reasons for not adopting constructive stance in mathematics?
4
votes
2answers
282 views

Did Structuralism influence the formulation of Category Theory?

Having only the a very cursory knowledge of Structuralism ( it's a movement generally held to have originated in linguistics, then moving on to philosophy & literature), there does appear to be ...
6
votes
6answers
2k views

Michael Spivak in “Calculus” asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by “exist”?

Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why ...
8
votes
6answers
2k views

Why accept the axiom of infinity?

According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but ...
5
votes
1answer
583 views

How do mathematical objects relate to the real world? (a little philosophy)

I am just going to give an example of what I mean using Skolem's Paradox. I don't want to get into Skolem;s Paradox itself or its "resolution." Skolem's showed that in first-order formulations of ...
26
votes
2answers
696 views

A few questions about intuitionistic mathematics

I have to write a paper on Intuitionism for my Philosophy of Science class and I'm struggling with a few concepts I have encountered in my self-study. The (intuitive) characterization of valid ...
4
votes
4answers
316 views

Are there any situations where you can only memorize rather than understand?

I realize that you should understand theorems, equations etc. rather than just memorizing them, but are there any circumstances where memorizing in necessary? (I have always considered math a logical ...
5
votes
4answers
2k views

Is McGee's counterexample to Modus Ponens accepted by the mathematical community?

In the mid 1980's Vann McGee proposed a counterexample to Modus Ponens: (a) If a Republicans will win the election, then if Reagan will not win, Anderson will win. (b) A Republican will win the ...
1
vote
1answer
255 views

Equivalence of sequences and subsets of natural numbers

For me, facts like the independence of the continuum hypotheses from ZFC cast a doubt on the "law of the excluded middle". (In this context, the doubt is that there might be no "final set theory" such ...