111–120 of 120 posts
> We cannot tell at a glance whether 9999999999999999 and 999999999999999 are the same. Off topic - some of us can, iOS underlines one of these for me, treating it as a phone number…
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The problem is that the definitions of in such footnotes would be full of terms that one also needs defined. For example here's the first few sentences from the Wikipedia page on Compact Sets: In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.[1] The idea is that a compact space has no "punctures" or "missing…
I'd like to see a series of sites or interactive ebooks that work as described below, which would address that problem. I can't actually build such a site or ebook because (1) I'm not sufficiently mathematically sophisticated and (2) my front end skills aren't up to it. Each one would be devoted to one interesting major mathematical theorem, such as the prime number theorem. The initial view would be a presentation o…
Automated theorem proverbs can probably solve this problem though..
I had a professor who derived Riemann metric from just area of triangle and limits over the course of a semester. So I see what you're getting at, but such books will probably he too long for most people to read anyway.
Isn’t this just taking a common sense concept and adapting it to the concepts and language of Turing machines and also topology? While yes it shows these are good tools, capable of confirming/proving these intuitions and showing exactly how these specific systems achieve that, is there ever any concern we are making too many redundant concepts? That may lead to confusion and obfuscate the search for novel ideas, whic…
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Next week on Show HN: I made a fractal font with unlimited scaling
The Minds from The Culture scifi series have something like this... The top level glyph is the base meaning and you can encode essentially infinite depth of meaning beyond that... Of course you have to go multidimensional at some point, so you can't really print it out beyond a few layers of resolution...
Perhaps the sense of "I" can be that base, which then allows the expansion of I, I, I (sense of I around I) which expands to in front, behind, etc.
Why do we need the property that X is compact? Say we define X (the square) to exclude the boundary: X = (0,1)x(0,1). This is no longer a compact set, but it seems silly for the proof to now fail. Is boundedness sufficient?
"I am assuming that the reader is familiar with the terms metric, metric space, topological space, and compact set." On behalf of the math-declined folks, I wish math articles had subtitles, references, and definitions built-in. I wonder if a browser extension for Greek letter definitions would be possible?
The problem is that the definitions of in such footnotes would be full of terms that one also needs defined. For example here's the first few sentences from the Wikipedia page on Compact Sets: In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.[1] The idea is that a compact space has no "punctures" or "missing…
[1] I don't know if this is still the case, but it certainly was in the late '00s.
As an aside, this is an interesting companion read:
"A pedagogical history of compactness" (https://arxiv.org/abs/1006.4131)
As a working mathematician, I can say that this kind of argument has become totally routine and, were I reading Turing's paper carefully, after seeing "epsilon" and "compact" I would think "makes sense" and move on. But, historically speaking, it's interesting to realize how recent the development of the abstract idea of compactness was when Turing was writing---the time between Frechet and Hausdorff's work on compactness (see the pedagogical history) and Turing is about the time between Google being founded and today.
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If the observation time is finite, and the time resolution of the observation is limited, then you'd still only have a finite number of distinguishable symbols, wouldn't you?
That's true if the time resolution is limited. Theoretically there could be no limit. It would not make for a very practical alphabet, but it would still be an unlimited alphabet.
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> here if anywhere :-D We can play a lot of games with "what is a symbol", but compactness pervades many of the models that we use to describe reality. The crux of the argument is not necessarily that the symbols themselves are compact as sets, but that the *space of possible descriptions* is compact. In the article, the space of descriptions is (compact) subsets of a (compact) two-dimensional space, which (delightfu…
The proof defines F(X) as the set of compact subsets of X (here the unit square). The compactness of F(X) follows from (necessary?) the compactness of its elements, so we need to find a topology where all our allowable symbols are compact, and this is not the standard topology if you allow the spiral. If you take another topology (equivalently space of symbol descriptions) then you again must show that this compactne…