Timeline for In what ways is physical intuition about mathematical objects non-rigorous?
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| Jun 19, 2014 at 15:16 | comment | added | Terry Tao | In any event, how certain could one be that failure of claim i and failure of claim j could not possibly be (completely or partially) statistically independent? I doubt that one's certainty on this issue could be anywhere close to $1 - 10^{-10^{100}}$. | |
| Jun 19, 2014 at 15:13 | comment | added | Terry Tao | One does not need full statistical independence here; even if only a tiny portion (say $10^{-10^{100}}$) of the event of failure for $i$ is independent of the event of failure for $j$, then there is still a strong likelihood of breakdown of the argument once $n$ exceeds, say, $10^{2 \times 10^{100}}$. For instance, an argument that somehow proceeds through an induction over all possible ensembles of an N-particle system, appealing to the second law of thermodynamics for each such ensemble, would be subject to this problem and be mathematically dubious. | |
| Jun 19, 2014 at 4:53 | comment | added | user21349 | 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, 2014 at 16:31 | comment | added | Terry Tao | If $n$ is large (e.g. larger than $10^{10^{100}}$), then it becomes essential in mathematics that each of the claims $P(i) \implies P(i+1)$ are known with $100\%$ certainty (again, in the internal sense of the mathematical reasoning model), as even a $10^{-10^{100}}$ failure rate for each of these steps will throw the derivation of $P(n)$ into serious doubt. This requirement of absolute certainty is particularly in effect for mathematical arguments involving countable or uncountable infinities; one simply cannot afford any failure probability at all, no matter how small. | |
| Jun 18, 2014 at 16:27 | comment | added | Terry Tao | I should perhaps clarify that in my post above, terms such as "certainty" and "probability" were meant to be interpreted internally within one's accepted mode of reasoning (whether physical or mathematical), and not with regards to the external degree of accuracy to which one can actually apply this reasoning. For instance, a common technique in mathematical reasoning is mathematical induction, in which a claim P(n) is deduced from an initial claim P(0) and the claims $P(i) \implies P(i+1)$ for $i=0,\dots,n-1$. (cont.) | |
| Jun 18, 2014 at 15:15 | comment | added | user21349 | 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, 2014 at 15:07 | comment | added | user21349 | 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. | |
| Jun 16, 2014 at 20:33 | history | edited | Kevin | CC BY-SA 3.0 |
Added $ signs.
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| Jun 13, 2010 at 10:02 | comment | added | Terry Tao | Ah, I was referring to the "offensive" side of cryptography (code-breaking) rather than the "defensive" side of cryptography (code design). The two sides of cryptography have opposite standards of rigour (and for good reason). I discuss this at google.com/buzz/114134834346472219368/FXab3PJgWhu/… | |
| Jun 13, 2010 at 5:36 | comment | added | Torsten Ekedahl | I think the example of cryptography is a little bit more complicated. Mathematicians working in cryptography are usually very careful to state as a hypothesis that certain problems are difficult (or just that there are difficult problems) much in the same way that certain results are conditional on the Riemann hypothesis. On the other hand physicists working in quantum cryptography seem to use "prove" in the sense of physics. Hence one sees statements to the effect that quantum cryptography is proved to be secure whereas RSA is not. | |
| Jun 13, 2010 at 2:06 | comment | added | The Mathemagician | Mathematics and physics are joined at the hip;they are both expressed using the same language (i.e. structures built from logical expressions and/or set theoretic constructs). But the driving motivation of each has become very different in the last 2 centuries. Mathematics is driven by logical consistency while physics needs empirical validation that's far less certain. While it is clear they draw from each other vigorously-particularly in algebra and geometry-they cannot be intechanged. | |
| Jun 12, 2010 at 17:56 | history | edited | Terry Tao | CC BY-SA 2.5 |
deleted 1 characters in body
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| Jun 12, 2010 at 17:48 | history | answered | Terry Tao | CC BY-SA 2.5 |