Questions involving philosophy of mathematics

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Big Topics in Mathematics [closed]

My question is as follows: It is now the year 2013 as we know it, and I'm wondering what the "big topics" in mathematics are. What fields are of utmost interest and foundation in the modern era? How ...
20
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5answers
644 views

What does it mean for a set to exist?

Is there a precise meaning of the word 'exist', what does it mean for a set to exist? And what does it mean for a set to 'not exist' ? And what is a set, what is the precise definition of a set?
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2answers
191 views

Gray's “Plato's Ghost” - a curious mistake

I am currently reading Jeremy Gray's "Plato's Ghost", and I run into the following passage (Chapter 5, page 332). The point is, it seems to me that it contains two very elementary mistakes that feel ...
4
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1answer
154 views

Does ZFC have an intended interpretation?

I know that PA has an intended interpretation, namely $\mathbb{N}$, and the usual axioms of the real line have an intended interpretation, namely $\mathbb{R}$. Does ZFC have an intended ...
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2answers
238 views

Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?

I've heard people make the argument that: $\mathsf{ZFC}$ suffices as a foundations of mathematics because almost all theorems in the mathematics literature can be proven using $\mathsf{ZFC}$, so ...
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6answers
957 views

What is this physicist saying?

I do not want to poison this forum with politics. But I want to understand, precisely, what is meant by the bolded statement. It is made by a physicist who used to work at Harvard regarding the ...
11
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3answers
354 views

What have been some of the most revolutionary philosophical shifts in perspective in mathematics?

Often times, great revolutions in mathematics come from shifts in philosophical perspective. The shift from extrinsic to intrinsic geometry yields manifolds (and much else). The shift in focus from ...
35
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8answers
3k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
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3answers
287 views

Quantum Mathematics?

As per my last question, this has less to do with cold, hard, and fast calculations and more to do with the interplay between mathematics and philosophy...but as armchair philosophers aren't as hard ...
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1answer
138 views

Intuitionism - is it fundamentally different than “ordinary” mathematics.

I have recently had a conversation with a person who considered intuitionism to be a valid alternative for the "usual" kind of mathematics. Clearly, intuitionism differs from the type of mathematical ...
2
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2answers
284 views

Are axioms and rules of inference interchangeable?

There is an equivalence between cellular automata and formal systems, you can code one into the other and vice versa. But in the the cellular automata (CA) the rules of inference are fixed and are ...
7
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3answers
184 views

Measure of how much information is lost in an implication

In an implication like $p \implies q$, is there some measure of how much information is lost in the implication? For example, consider the following implications, where $x \in \{0,1,\ldots,9\}$: ...
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2answers
275 views

Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?

My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
35
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8answers
4k views

Infinite sets don't exist!?

Has anyone read this article? Set theory This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his ...
14
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1answer
308 views

What lessons have mathematicians drawn from the existence of non-standard models?

So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily ...
2
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3answers
194 views

Subsets as non-mathematical objects?

I think of mathematical objects as individual things that exist by their own (either abstractly or concretely) and can be represented mathematically. When thinking of subsets, I'm in doubt if ...
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2answers
147 views

Just a thought… defining “competition”?

Let's think about galaxies and animals. At first, they seem completely different. But their behavior seems to be governed by (or at least arise from) the same rules. Think about competition. ...
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5answers
1k views

What's the best way to measure mathematical ability?

Very soft question I admit, but it's something that's been bothering me for a while. I've been thinking that being self taught has the problem of accreditation. You can't evaluate a mathematician ...
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13answers
3k views

Is there such a thing as proof by example (not counter example)

Is there such a logical thing as proof by example? I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right. This works and is ...
6
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3answers
194 views

Mathematical Limitations of Computer Experiments

One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never ...
2
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1answer
186 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
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1answer
211 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
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1answer
152 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
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1answer
111 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.
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2answers
85 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 ...
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1answer
138 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, ...
15
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1answer
347 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 ...
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2answers
202 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?
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2answers
217 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 ...
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0answers
105 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?
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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 ...
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10answers
815 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
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5answers
474 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 ...
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1answer
282 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 ...
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4answers
258 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 ...
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5answers
537 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
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0answers
235 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 ...
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3answers
447 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 ...
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5answers
1k 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 ...
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1answer
123 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
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1answer
239 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 ...
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4answers
278 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
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2answers
405 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
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1answer
174 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
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4answers
677 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 ...
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4answers
329 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?
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2answers
179 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 ...
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4answers
585 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 ...
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4answers
6k 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 ...
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2answers
119 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 ...