Questions involving philosophy of mathematics
205
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 ...
97
votes
19answers
6k 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 ...
42
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 ...
40
votes
5answers
2k views
In what sense are math axioms true?
Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers.
The kid asks: why?
Well, it's an axiom. It's called commutativity (which is not even true for most groups).
How do I ...
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 ...
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 ...
34
votes
17answers
3k views
What's the goal of mathematics?
Are we just trying to prove every theorem or find theories which lead to a lot of creativity or what?
I've already read G. H. Hardy Apology but I didn't get an answer from it.
33
votes
12answers
4k views
I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]
I didn't know how to phrase the question properly so I am going to explain how this came about.
I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
30
votes
3answers
614 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 ...
29
votes
16answers
2k views
Non-Scientific questions solved by mathematics
I have a general question about the applications of mathematics.
What are some applications of mathematics that are not scientific, perhaps maybe literary or philosophical, or political.
I am ...
22
votes
2answers
493 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 ...
21
votes
6answers
1k views
If all sets were finite, how could the real numbers be defined?
An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who ...
20
votes
6answers
1k views
What are natural numbers?
What are the natural numbers?
Is it a valid question at all?
My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
20
votes
5answers
713 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?
(PD: I'm not religious)
20
votes
1answer
318 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 ...
20
votes
3answers
596 views
Rejecting infinity
I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...
19
votes
9answers
2k views
Good books on Philosophy of Mathematics
Where can I learn more about the implications, meta discussions, history and the foundations of mathematics?
Is Russell's Introduction to Mathematical Philosophy a good start?
19
votes
4answers
339 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, ...
17
votes
1answer
1k views
$e^{e^{e^{79}}}$ and ultrafinitism
I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. ...
16
votes
1answer
605 views
What did Gauss think about infinity?
I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
13
votes
4answers
309 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?
13
votes
5answers
467 views
Definition of definition
I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
13
votes
1answer
226 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 ...
12
votes
7answers
894 views
Reference request: is mathematics discovered or created?
I have to write a short monograph as an assignment for a course on the philosophy of science. Being a math student, of course I want to opt for something math-related. After some initial ideas which ...
12
votes
3answers
292 views
Is there any difference between a math invention and a math discovery? [closed]
From wikipekia:
The calculus controversy was an argument between 17th-century
mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented
in part by their disciples and associates – ...
11
votes
6answers
1k views
Why do statements which appear elementary have complicated proofs?
The motivation for this question is : http://math.stackexchange.com/questions/4066/rationals-of-the-form-fracpq-where-p-q-are-primes-in-a-b and some other problems in Mathematics which looks as if ...
11
votes
9answers
853 views
Problems that are largely believed to be true, but are unresolved
Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification?
It seems that Goldbach should be true, but this is based on heuristic ...
11
votes
6answers
970 views
What philosophical consequence of Goedel's incompleteness theorems?
I want to write a philosophical essay centered about Goedel's incompleteness theorem. However I cannot find any real philosophical consequences that I can write more than half a page about. I read the ...
10
votes
2answers
176 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 ...
10
votes
1answer
264 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
5answers
813 views
The status of high school geometry
Okay, so we've all seen Euclidean geometry in primary and high school. Back then, I really thought of points as indivisible entities in space and lines as 'breadthless lengths'. As far as I could ...
9
votes
6answers
812 views
What mathematical questions or areas have philosophical implications outside of mathematics?
Please list both the problem/area and justify why it is important philosophically. This question doesn't cover questions that are only important within the philosophy of mathematics itself.
9
votes
4answers
792 views
Why does Benford's Law (or Zipf's Law) hold?
Both Benford's Law (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and Zipf's Law (given a corpus of natural ...
9
votes
0answers
113 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 ...
8
votes
3answers
486 views
Difference between undecidable statements in set-theory and number theory?
Do all statements about the integers have a definite truth value?
For instance: Goodstein's theorem is clearly true, otherwise we could find a finite counterexample thus it would be possible to ...
8
votes
6answers
490 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 ...
8
votes
3answers
253 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 ...
8
votes
4answers
365 views
Consequences of solving the Halting problem
What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math?
Would there be any mathematical problems left to solve?
What ...
8
votes
5answers
532 views
Time in Mathematics
I claim that it is commonly believed that Mathematical objects can be seen as genuinely static, with no "Platonic" time in which they do genuinely evolve.
Nevertheless time has its place in ...
8
votes
1answer
237 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 ...
7
votes
5answers
951 views
Common misconceptions about math
YARFMO (Yet another reposting from Mathoverflow) ;-)
The more you know about math the more you find conceptions previously thought correct to be false:
1.) math is not as exact as many believe - in ...
7
votes
5answers
435 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 ...
7
votes
1answer
238 views
When can we say that a theorem has been proven?
I'm taking a Data Structures and Algorithms course for a CS program. The introductory material was all mathematics, mostly a series of formulas that we are to remember. I can work through the formulas ...
7
votes
2answers
156 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 ...
7
votes
4answers
315 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 ...
7
votes
4answers
326 views
Are the computable reals finitary?
In the comment thread of an answer, I said:
The computable numbers are based on the intuitionistic continuum, and are not finitary.
To which T.. replied:
Computable numbers are not based on ...
7
votes
2answers
367 views
How many different proofs can a theorem have?
I notice some problems has many different proofs, do all theorems have multiple proofs, is there some theorems which has only 1 way to prove it? $n$ ways? infinite?
7
votes
4answers
370 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 ...
7
votes
2answers
138 views
Literature on general paradox?
I suppose this one teeters on the edge of un-mathematical, but here it goes...
I've been on something of a logic binge lately and have (surprise, surprise!) especially been interested in the results ...
7
votes
3answers
130 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 $x \in \{0,1,\ldots,9\}$:
...
