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visits | member for | 2 years, 6 months |
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stats | profile views | 561 |
Aug 7 |
answered | Papers that debunk common myths in the history of mathematics |
Jul 28 |
awarded | Nice Answer |
Jul 24 |
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What makes four dimensions special?
3+1 is the smallest dimensionality in which general relativity is interesting. In 2+1 dimensions gravitational fields don't propagate and the theory becomes purely topological. |
Jul 19 |
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Negative impact of wrong or non-rigorous proofs
There are probably some good examples from premodern Euclidean geometry, such as naive proofs relating to infinities, or proofs of the parallel postulate. |
Jul 19 |
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Negative impact of wrong or non-rigorous proofs
@quid: It answers the Newton-related part of question in the negative: no lack of rigor, no incorrect proofs, no damage. |
Jul 19 |
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Negative impact of wrong or non-rigorous proofs
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Jul 19 |
answered | Negative impact of wrong or non-rigorous proofs |
Jul 17 |
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How does one justify funding for mathematics research?
It's enjoyable reading the answers, but MO is the worst possible place to ask the question. If there was a plumbers.SE, people there would tell you that plumbing was worthy of the highest social status and vast taxpayer-provided subsidies. After all, good plumbing was what made the Roman Empire great (aqueducts, public baths), and bad plumbing may also have been what brought it down (lead pipes). |
Jul 16 |
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Mathematica package for supergravity and string theory
You may also be interested in the CAS cadabra and in the ctensor package for the CAS Maxima. These are all free and open-source. |
Jul 12 |
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What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
related: math.stackexchange.com/questions/865559/… |
Jul 12 |
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What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
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Jul 12 |
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What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
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Jul 12 |
answered | What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail? |
Jul 12 |
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What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
@AndrejBauer: but when you say that dy/dx refers to the standard part, you're dragging in Robinson's non-standard analysis Check out the link in my earlier comment, p. 10. Fermat and Leibniz had a notation, ${}_{\ulcorner\!\urcorner}$, and a term, "adequality," that expressed essentially the same notion as Robinson's standard part (subject to the limitations of the development of mathematics at that time). The idea is much more generic than NSA. |
Jul 12 |
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What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
@DonuArapura: Roger Penrose introduced abstract index notation, which combines the expressiveness of index notation with the coordinate independence of coordinate-free notation. |
Jul 11 |
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What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
@AndrejBauer: There is nothing wrong with dy/dx; you just have to understand the implication that it's referring to the standard part. This was in fact understood pretty well in Leibniz's lifetime: arxiv.org/abs/1202.4153 . In your example of $\int x^2dx=x^3/3+C$, you can think of $x$ as the identity function rather than as a bound variable, and it makes perfect sense. There is nothing broken about the Leibniz notation. The Leibniz notation has many wonderful advantages, and that's why it caught on quickly and has been used universally ever since. |
Jun 25 |
answered | What to read for many-body problems in 3D Schrodinger equation |
Jun 19 |
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In what ways is physical intuition about mathematical objects non-rigorous?
I think we're in agreement that certainty and probability are fuzzy and/or discipline-specific terms. The argument about induction seems weak to me. It seems to depend on an assumption that the falsehood of claim $i$ is statistically independent of the falsehood of claim $j$ for $i\ne j$. I find that implausible. |
Jun 18 |
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In what ways is physical intuition about mathematical objects non-rigorous?
In physics, it's OK if Maxwell's demon [...] steps in to ruin everything $10^{−100}$ of the time; but this is unacceptable by the standards of rigorous mathematics. The probability is more like $P=10^{-10^{100}}$. This is a far better certainty than anything in mathematics. $P$ is much smaller than the probability that Nelson will fix up his proof that Peano arithmetic is inconsistent. $P$ is also much smaller than the probability that a proof assistant like Coq will make an error due to a software bug. More importantly, there is no way, even in principle, to make these probabilities $<P$. |
Jun 18 |
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In what ways is physical intuition about mathematical objects non-rigorous?
Mathematics [...] has the luxury of insisting on near-100% certainty. The question is about rigor, not certainty. Physicists submit themselves to the rigor of needing to make correct predictions about the outcomes of experiments, and this is simply a different kind of rigor than the kind involved in verifying a preexisting mathematical proof (which is only a small portion of what's done in the profession of mathematics). Rigor and certainty don't go hand in hand. We can prove rigorously from the axioms of quantum mechanics that there is uncertainty about when a nucleus will decay. |