Questions involving philosophy of mathematics
105
votes
20answers
7k 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 ...
13
votes
5answers
337 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?
5
votes
3answers
107 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
99 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 ...
5
votes
3answers
220 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 ...
1
vote
1answer
450 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 ...
4
votes
1answer
99 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 ...
10
votes
2answers
180 views
“Optical Illusion” in 4D
(Apologies in advance for the wordiness - not very mathematical, I know!)
I open my book of Escher optical illusions and look at the 2D page. "Aha!" my brain says - "That image doesn't make sense ...
0
votes
0answers
54 views
Why geometry is so important part of math for physics? [closed]
I mean geometry in broad (with differential geometry, topology, etc.).
37
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 ...
44
votes
6answers
1k 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 ...
30
votes
3answers
658 views
What are examples of unexpected algebraic numbers of high degree occured in some math problems?
Recently I asked a question about a possible transcendence of the number ...
4
votes
0answers
58 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 ...
20
votes
4answers
376 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, ...
20
votes
1answer
335 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 ...
4
votes
2answers
141 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 ...
7
votes
3answers
134 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 $x \in \{0,1,\ldots,9\}$:
...
-2
votes
1answer
175 views
What does it mean that a number exists? [closed]
The word exists is a very culturally loaded word. Yet mathematicians use it all the time. My answer proposing that the statement that there "exist a number Q such that Q*Q=2", only means that we can ...
206
votes
33answers
17k views
Do complex numbers really exist?
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
6
votes
6answers
864 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 ...
1
vote
2answers
52 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 ∞} ...
0
votes
2answers
111 views
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 ...
3
votes
5answers
362 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
0answers
64 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 ...
3
votes
0answers
89 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 ...
4
votes
2answers
147 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 ...
8
votes
3answers
254 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 ...
4
votes
4answers
124 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?
3
votes
3answers
208 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 ...
6
votes
1answer
218 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 ...
3
votes
2answers
85 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 ...
2
votes
1answer
76 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 ...
7
votes
2answers
158 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 ...
9
votes
0answers
121 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
88 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
107 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. ...
5
votes
3answers
117 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
128 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 ...
6
votes
1answer
128 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:
...
3
votes
0answers
106 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 ...
2
votes
1answer
75 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?$$
Related, but distinct: Allowed probabilities under frequentism.
2
votes
0answers
65 views
Allowed probabilities under frequentism
Am I right to assume that under the frequentist interpretation of probability,* the set of allowed probabilities isn't $$\left[0,1\right],$$
but rather is
...
2
votes
1answer
152 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 ...
1
vote
0answers
67 views
A characterization of probability theory [closed]
Is the following a good characterization of probability theory:
The study of measure spaces space $(X, \mathcal{B}, \mu)$ such that $\mu(X) = 1$, with random variables being measurable functions $f: ...
3
votes
1answer
118 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, ...
13
votes
1answer
228 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
122 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 ...
1
vote
2answers
186 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?
39
votes
4answers
2k views
What is “ultrafinitism” and why do people believe it?
I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone ...
5
votes
2answers
163 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 ...
