The meta-math tag has no wiki summary.
42
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6answers
2k views
Why are all the interesting constants so small?
A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is ...
30
votes
2answers
832 views
What is your method?
Many young, and not so young, mathematicians struggle with how to spend their time. Perhaps this is due to the 90%-10% rule for mathematical insight: 90 pages of work yield only 10 pages of useful ...
27
votes
5answers
1k views
Why is it considered unlikely that there could be a contradiction in ZF/ZFC?
EDIT:
No answer addresses the "bottleneck" question. It's not surprising to me because the question is vague. But I would like to know whether that is indeed the reason, or perhaps something else. ...
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 ...
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. ...
11
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2answers
504 views
How rare is it that a theorem with published proof turns out to be wrong?
There is a story I read about tiling the plane with convex pentagons.
You can read about it in this article on pages 1 and 2.
Summary of the story:
A guy showed in his doctorate work all classes of ...
10
votes
6answers
2k views
Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?
OK, maybe the title is exaggerated, but is it true that the rest of math is just "good enough", or a good approximation of absolute truth - like Newtonian physics compared to general relativity? How ...
9
votes
6answers
682 views
Geometric proof of existence of irrational numbers.
It is easy, using only straightedge and compass, to construct irrational lengths, is there a way to prove, using only straightedge and compass, that there are constructible lengths which are ...
9
votes
2answers
213 views
Are there formal systems that can not be proved to be complete or incomplete?
I'm reading GEB and was thinking about this. Are there any formal systems where the proof of their completeness/incompleteness is unprovable?
This question could go ad infinitum. Could there be any ...
8
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6answers
1k views
Is It True that We Can Never Be Sure of Validity of a Mathematical Proof?
The reason I ask this is because difficult mathematical proofs are just not plain self-evident. You would need a few years of intensive study before you can get to the point of understanding the ...
8
votes
4answers
274 views
Clarification of a remark of J. Steel on the independence of Goldbach from ZFC
On page 424 of the following paper:
S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, ``Does Mathematics Need New Axioms?'' The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. ...
8
votes
1answer
383 views
How to address mistakes in published papers?
I have recently discovered some mistakes in a published maths article. I have contacted the author pointing out politely my concerns, but I got no specific answer, just a "polite" one, that the ...
7
votes
5answers
952 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
3answers
202 views
Is there a way to prove that a theorem has no elementary proof? Or to prove that something may have no proof?
Recently I was trying to prove something, more or less elementarily, but eventually started going in circles. My prof said that the proof involves mathematical tools that I've not seen yet, and that ...
7
votes
1answer
712 views
Fiction “Division by Zero” By Ted Chiang
Fiction "Division by Zero" By Ted Chiang
I read the fiction story "Division by Zero" By Ted Chiang
My interpretation is the character finds a proof that arithmetic is inconsistent.
Is there a ...
6
votes
3answers
250 views
Meaning and example(s) of Qiaochu's quote.
I happen to come across this page http://math.uchicago.edu/~chonoles/quotations.html which contains some beautiful quotes by various mathematicians and I came across Qiaochu's quote as claimed by the ...
6
votes
3answers
273 views
Proposition vs Theorem
What is the distinction between a proposition and a theorem? How do people decide which of the 2 to use in, say, textbooks? Somehow I think proposition sounds less serious... Thanks.
6
votes
3answers
130 views
If $T$ proves $\operatorname{Con}(ZFC)$, is $T$ at least as strong as set theory?
I am looking for either a proof of counterexample of this:
Lemma: Let $\pi$ be a faithful interpretation of $PA$ into $ZFC$, and let $PA'$ be the image of $PA$ under $\pi$. If there is a $T$ with ...
6
votes
1answer
119 views
Are there any strongly axiomatizable logics that are not compact?
I mean here a logic in the sense of a language and semantics. By strongly axiomatizable I mean strongly sound and strongly complete. So I'm basically asking if there is a particular deductive system ...
6
votes
1answer
248 views
Formalizing metamathematics
I am reading historical/philosophical stuff on the concept of "metamathematics" and am by now quite confused. Several questions emerged, but they are probably somehow confused and interrelated, I ...
5
votes
3answers
269 views
Are “axioms” in topology theory really axioms?
If I understand correctly, axioms are those statements that we assume to be true, instead of proving to be true.
I have seen that in topology theory, various axioms of countability and separation ...
5
votes
2answers
529 views
Definition of “non-constructive proof”
I was wondering if it is possible to define exactly what a non-constructive (nc) proof is. I have often seen the concept associated with the use of principles such as the axiom of choice or the law of ...
5
votes
1answer
269 views
Mathematical structures
Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
4
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6answers
628 views
Why some people don't like proofs by contradiction [duplicate]
Possible Duplicate:
Are the “proofs by contradiction” weaker than other proofs?
I have been active on this site for two months and on a few occasions I noticed that some people ...
4
votes
5answers
514 views
Is mathematics the only language that is not subject of interpretation?
Do you know any other "language" that is used by people except mathematics and is not subject of interpretation? By subject of interpratation I mean e.g. that 1 000 000 people will undertand that 1 + ...
4
votes
3answers
145 views
Has anyone ever tried to develop a theory based on a negation of a commonly believed conjecture?
I know that plenty of theorems have been published assuming the Riemann hypothesis to be true. I understand that the main goal of such research is to have a theory ready when someone finally proves ...
4
votes
0answers
68 views
An “internal” condition on $T$ so that for the standard provability predicate, $T$ proves $\text{Pf}(\underline S)$ implies $T$ proves $S$?
This is probably quite basic, but I'd like to make sure I got this right. Regarding the proof of Goedel's first incompleteness theorem, say that we have $T$ containing $PA$ effectively axiomatizable ...
3
votes
2answers
298 views
What branch of mathematics is most needed in the industry or how one can make living with mathematics (apart from teaching)?
If you learn carpentry or programming you have the clear options of becoming a carpenter or programmer. But what if you learn mathematics?
I know that there are some financial institutions out there, ...
3
votes
1answer
145 views
What are metatheory, metalanguage and meta-…
I have been reading the Wiki articles for metatheory and metalanguage, but not sure if I have understood what they are about. Some accessible examples may help clarify a bit, I guess.
Do metatheory ...
3
votes
2answers
72 views
Does a “solution map” system like this already exist?
I was doing this simple Calc 1 problem and it took me forever to get it right and it was embarrassing. I could see that the problem was easy but I just couldn't 'see' what I was doing. I couldn't ...
3
votes
0answers
554 views
New branches of math? [closed]
I have been wondering if math would be more enjoyable, if one was able to start a new field and come up with all the definitions, methods, etc. rather than starting where someone else ended. ...
2
votes
2answers
699 views
Is Gödel's theorem invalid? [closed]
Right now I've skim through Gödel's theorem is invalid by Diego Saá on arXiv (freely available).
As it seems very plausible, I ask for any references and scrutinizations of the paper.
2
votes
2answers
178 views
Is there a way of defining the notion of a variable mathematically?
I know that the notion of "set" is one that cannot be defined mathematically since it is the fundamental data type that is used to define everything else (and the definition which says that "sets" are ...
2
votes
1answer
191 views
Truth and undecidability
I believe this is more of a philosophical question.
Given a consistent theory T and a statement S independent of T. Can S be true or false in T? (I don't see any contradiction with that)
I read that ...
2
votes
1answer
75 views
question about Godel numbering
I have a question about Godel numbering, it is trivial but I would like to know how can you know the length of an expression through its Godel number. ¿?
I think you can use a recursive function but ...
2
votes
1answer
98 views
Separation of mathematics and metamathematics
I recall reading that it's important to separate mathematics and metamathematics. What exactly does this mean, and why is it so?
I understand that this question may make no sense without more ...
2
votes
1answer
159 views
Mathematical structures and signature
From Wikipedia:
In mathematics, a structure on a set, or more generally a type,
consists of additional mathematical objects that in some manner attach
(or relate) to the set, making it easier ...
2
votes
2answers
130 views
A theorem about inductive inference
In the book 'Introduction of the theory of Statistics' by Mood,Graybill,Boes (third edition)on page 220 (Chapter 6 on Sampling) you can read:
'Inductive inference is well known to be a hazardous ...
2
votes
0answers
201 views
Statements about logic (=“metalogic”(??))
Sometimes there are statements about logic e.g. "That's not logical" and I can neither prove nor disprove a statement about logic with no definition for logic itself. It's just a negation and it's ...
1
vote
1answer
68 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 ...
1
vote
2answers
197 views
The mathematics of mathematical knowledge
It's been many years since I did any real mathematics but last night after pondering the process involved in my mathematical journey I had an idea about the abstraction of how mathematical analysis ...
1
vote
1answer
178 views
Making meaning of mathematical “bridges”
I apologize for posting such an untechnical question, but with responses it could surely be posed in a better form.
I'm a math noob, but I've seen (as we all have) a few examples of "connections" ...
