Questions tagged [soft-question]

For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

For questions whose answers cannot be objectively evaluated as correct or incorrect, but which are still relevant to this site.

11201 questions
168
votes
9 answers

Why do people use "it is easy to prove"?

Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the phrase "it is easy to see/prove/verify/..." in the…
oleksii
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166
votes
29 answers

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan. While it's a good book,…
165
votes
20 answers

Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your contributions, in advance!
162
votes
22 answers

Examples of mathematical discoveries which were kept as a secret

There could be several personal, social, philosophical and even political reasons to keep a mathematical discovery as a secret. For example it is completely expected that if some mathematician find a proof of $P=NP$, he is not allowed by the…
user180918
161
votes
33 answers

Can you provide me historical examples of pure mathematics becoming "useful"?

I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because it has no practical utility, but I guess that…
Red Banana
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159
votes
20 answers

How to distinguish between walking on a sphere and walking on a torus?

Imagine that you're a flatlander walking in your world. How could you be able to distinguish between your world being a sphere versus a torus? I can't see the difference from this point of view. If you are interested, this question arose while I was…
Julien__
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159
votes
31 answers

Stopping the "Will I need this for the test" question

I am a college professor in the American education system and find that the major concern of my students is trying to determine the specific techniques or problems which I will ask on the exam. This is the typical "will this be on the test?"…
Wintermute
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159
votes
33 answers

What are the most overpowered theorems in mathematics?

What are the most overpowered theorems in mathematics? By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. I'm looking for the biggest guns a research…
Samuel Handwich
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159
votes
8 answers

Intuition of the meaning of homology groups

I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible I would prefer it if this could be kept…
158
votes
7 answers

Is Apple ipad / tablet good for mathematics students?

I am a math student. I'd like to find out if tablets (iPads, Galaxy Note 10.1) are worth the cost. How good are tablets for the purposes of reading textbooks as PDF and writing mathematics with a stylus? For writing math in TeX, it looked like the…
T. Webster
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157
votes
20 answers

Are there any open mathematical puzzles?

Are there any (mathematical) puzzles that are still unresolved? I only mean questions that are accessible to and understandable by the complete layman and which have not been solved, despite serious efforts, by mathematicians (or laymen for that…
Řídící
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157
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8 answers

What's the point in being a "skeptical" learner

I have a big problem: When I read any mathematical text I'm very skeptical. I feel the need to check every detail of proofs and I ask myself very dumb questions like the following: "is the map well defined?", "is the definition independent from the…
Dubious
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156
votes
16 answers

Alternative notation for exponents, logs and roots?

If we have $$ x^y = z $$ then we know that $$ \sqrt[y]{z} = x $$ and $$ \log_x{z} = y .$$ As a visually-oriented person I have often been dismayed that the symbols for these three operators look nothing like one another, even though they all…
friedo
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154
votes
16 answers

What's new in higher dimensions?

This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3". What I am wondering about is what new geometrical phenomena are there in higher dimensions. When I say new I mean phenomena…
Chequez
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154
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25 answers

Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a function $f$: $f^{-1}(x)$ can be an inverse and a…
Frank Vel
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