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Results tagged with intuition
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user 290
Questions asking for the intuition behind some definition, conjecture, proof etc. In other words, questions designed to improve or to acquire understanding on a conceptual or intuitive level, as opposed to on a technical or formal level. When asking such a question it can be helpful to include a rough description of ones understanding of the subject at hand (on a technical level).
3
votes
Accepted
Explaining a comment: Difference between a transformation of points and a transformation of ...
Let $V$ be a finite-dimensional real vector space. Choosing a basis of $V$ amounts to giving an isomorphism $\phi : \mathbb{R}^n \to V$. Changing basis amounts to hitting $\mathbb{R}^n$ with an automo …
- 118k
4
votes
Most helpful heuristic?
A sort-of heuristic in combinatorics is that if you can't figure out what to do with a set, take the free abelian group / vector space on that set and work with linear transformations instead of funct …
3
votes
Intuitive Example of a Jacobson Radical
The intuition I have about the nilradical (and by extension, the Jacobson radical) is that it measures how far R is from behaving like the ring of functions on a space. …
- 118k
17
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3
answers
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What is a reasonable finitary analogue of the statement that harmonic functions are smooth?
In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way …
- 118k
6
votes
Why is addition of observables in quantum mechanics commutative?
This story cannot possibly be correct as written, and I agree with Fabian Besnard about why:
When you say "We can now define a sum and a product of observables. These are obtained by performing the t …
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32
votes
What is a coalgebra intuitively?
Coalgebras appear naturally in combinatorics as describing ways one can decompose objects into other objects of the same type. For example, the coalgebra structure on $k[x]$ given by
$$x^n \mapsto \s …
8
votes
Surprising and Useful Physical Intuition for Mathematical Objects
There are several examples at the number theory and physics archive. To get you started let me mention the statistical-mechanical interpretation of the Riemann zeta function as the partition function …
15
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Surprising and Useful Physical Intuition for Mathematical Objects
The physical intuition behind the orbit method comes from the notion of quantizing a classical system: the rough idea, as I understand it, is that orbits of the action of $G$ on $\mathfrak{g}^{\ast}$ should …
2
votes
Abstract nonsense versions of "combinatorial" group theory questions
Philosophically I think of the Sylow theorems as a statement about "localization of groups" at a prime. For abelian groups one can make this precise, since abelian groups, as Z-modules, can be realiz …
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15
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Why the Dold-Thom theorem?
This won't involve any geometry, but here is a model-independent description of the situation as I understand it. I will not prove anything. The very short summary is that
The infinite symmetric …
- 118k
108
votes
10
answers
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What is (co)homology, and how does a beginner gain intuition about it?
What's the intuition behind the definition of the boundary operator in simplicial homology? In what sense does homology count holes? …
10
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Proofs by induction
Let me see if I can address what I think is the underlying question. Probably most of the proofs by induction you've seen have been of the form "show that this identity $P(n)$ holds," where you are g …
- 118k
4
votes
Heuristic explanation of why we lose projectives in sheaves.
We can turn the question around to ask: why do we have projectives in module categories? One answer is that we know we have a plentiful supply of projectives because free modules are projective. Abstr …
- 118k
14
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Examples of using physical intuition to solve math problems
Mark Levi's book The Mathematical Mechanic is full of elementary and beautiful examples of this kind. Some examples are also given in this blog post by Yan Zhang.
A classic example is a "proof" that …
50
votes
6
answers
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Intuition for the last step in Serre's proof of the three-squares theorem
Bjorn Poonen, after presenting this proof in class, remarked that he had no intuition for why this should work. Does anyone have a reply? …
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