Questions involving philosophy of mathematics

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6
votes
3answers
159 views

There is concept of finite sets that can have only one “interpretation”?

In our mind we have a naive idea of what a set is, and in nature we can only observe something that behave like a finite set, ZFC (or set theories in general) tries to catch these properties in ...
3
votes
2answers
261 views

Why demonstrations are important in mathematics? [closed]

Good evening, I'm studying math and would like to know how important are mathematical proofs in the world and particularly in a school of mathematics Thanks for your help
3
votes
2answers
139 views

The consistency of PA is falsifiable. Can the same be said of its soundness?

The statement that 'PA is consistent' is absolutely falsifiable (a term I just made up), in the sense that if it is false, then we can demonstrate its falseness independently of any metatheory (just ...
6
votes
3answers
1k views

What would be the immediate implications of a formula for prime numbers?

What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
4
votes
1answer
313 views

Binary vs. Ternary Goldbach Conjecture

Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current ...
78
votes
9answers
4k views

How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
42
votes
3answers
997 views
6
votes
2answers
131 views

Axiom of Choice-esque argument to show that a proof of a statement exists without actually giving a proof

What if the set of all well-formed statements in ZFC formed a kind of pseudo-category where a morphism f between objects A, B represented a formal proof that A implied B? What if that category could ...
26
votes
4answers
1k views

Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
31
votes
1answer
665 views

What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions?

I got the number $$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$ in the ...
3
votes
2answers
193 views

Truth of Fundamental Theorem of Arithmetic beyond some large number

Let $n$ be a ridiculously large number, e.g., $$\displaystyle23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23}}}}}}}}}}}}+5$$ which cannot be explicitly written down provided the size of the ...
3
votes
2answers
737 views

Philosophy of a Mathematician.

Introduction I don't study Mathematics at university, and probably I don't have any chances to have a little understanding of what mathematics in all its aspects. But I love to find structures and ...
151
votes
23answers
12k views

Is mathematics one big tautology?

Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties ...
1
vote
2answers
507 views

Reinterpreting improper integrals that require Cauchy principal value to be defined

This question concerns the Cauchy principal value. Consider the improper integral $$\int_{-∞}^{∞}\frac{1+x}{1+x^2}dx$$ which is divergent, and then its Cauchy principal value $$\lim_{u \to ∞} ...
20
votes
5answers
693 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?
4
votes
2answers
194 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
votes
1answer
162 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 ...
7
votes
2answers
239 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 ...
7
votes
6answers
967 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
votes
3answers
369 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
votes
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 ...
5
votes
3answers
297 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 ...
2
votes
1answer
160 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
votes
2answers
330 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
votes
3answers
195 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 $x \in \{0,1,\ldots,9\}$: ...
6
votes
2answers
290 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 ...
36
votes
8answers
5k 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
votes
1answer
315 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
votes
3answers
212 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 ...
1
vote
2answers
150 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. ...
27
votes
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 ...
40
votes
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
votes
3answers
200 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
votes
1answer
192 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
225 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
154 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
87 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, ...
16
votes
1answer
384 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
230 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
106 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
904 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
482 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
292 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
545 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
310 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 ...