All Questions
Tagged with philosophy real-analysis
23
questions
2
votes
1
answer
134
views
Can the fundamental theorems of real analysis be proven/developed without proof by contradiction?
I've been reading about philosophical debates between mathematicians, and some seemed to reject the ideas of real analysis (such as the extreme value theorem) based on a school called "...
- 59
16
votes
13
answers
3k
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Does the epsilon-delta definition of limits truly capture our intuitive understanding of limits?
I've been delving into the concept of limits and the Epsilon-Delta definition. The most basic definition, as I understand it, states that for every real number $\epsilon \gt 0$, there exists a real ...
- 287
2
votes
1
answer
84
views
Real world relevance of uncountability of R [duplicate]
This may seem like a naive/stupid philosophical question so I am prepared to get destroyed here but what is the real world relevance of the uncountability of the real number system? I understand that ...
- 85
3
votes
0
answers
95
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links between real and complex analysis
I am new to analysis and just starting to appreciate it. So far I have encountered 3 what seem like fundamental links between the cases of $\mathbb{R}$ and $\mathbb{C}$:
The Cauchy-Riemann equations
...
- 323
1
vote
1
answer
78
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Lakatos on continuity and invariance to rotation
On page 159, note 21, of Cauchy and the continuum, Imre Lakatos writes: "The modern definition of continuity [the $\epsilon-\delta$ definition] is strongly counter-intuitive, e.g. it is not ...
- 436
51
votes
7
answers
8k
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How can someone reject a math result if everything has to be proved?
I'm reading a book on axiomatic set theory, classic Set Theory: For Guided Independent Study, and at the beginning of chapter 4 it says:
So far in this book we have given the impression that sets are ...
- 929
-2
votes
1
answer
79
views
the validity of (ε, δ)-definition of limit
People use this definition by constructing δ for arbitrarily small ε, or proving that for some ε, δ does not exist.
So my question is:
Is there a function that one can not construct or disprove the ...
- 9
3
votes
2
answers
439
views
Is there any "real" number that may not actually be a real because we haven't found its Dedekind cut?
I just watched a video that shows how real numbers are constructed using Dedekind cuts, and what I understood was that a real number is a subset of Q which, among a few other conditions, contains no ...
- 971
1
vote
0
answers
46
views
The use of square in maths
The variance of a random variable is defined as $E[(X-E[X])^2]$.
In machine learning and linear regressions, loss is sometimes calculated with the squared error.
In both cases, the main function of ...
- 227
23
votes
4
answers
5k
views
Why did mathematician construct extended real number systems, $\mathbb R \cup\{+\infty,-\infty\}$?
I know some properties cannot be defined with the real number system such as supremum of an unbounded set. but I want to know the philosophy behind this construction (extended real number system ($\...
- 657
-5
votes
1
answer
152
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What finite results cannot avoid mentioning the reals? [closed]
I am interested in theorems whose. . .
. . . hypotheses and conclusions are 'discrete' and mention only rational numbers.
. . . most natural proofs invariably require an understanding of the ...
- 10.4k
8
votes
4
answers
1k
views
What is... A Parsimonious History?
Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
- 43.8k
-1
votes
1
answer
96
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A question about the real line and the Dirichlet function.
Though the graph of the Dirichlet function is non-drawable, I think if we have to draw it in some informal way then it will be two complete lines (instead of isolated points).
Here's my reasoning: ...
- 8,656
1
vote
10
answers
5k
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Does it make any sense to prove $0.999\ldots=1$?
I have read this post which contains many proofs of $0.999\ldots=1$.
Background
The main motivation of the question was philosophical and not mathematical. If you read the next section of the post ...
user170039
1
vote
0
answers
532
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Computability, Continuity and Constructivism
Triggered by an IMO extremely interesting question & Mathematics Stack Exchange,
asked by Dal:
Computability and continuous real functions
And a link in one of the comments that could have ...
- 17.2k