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emacs users: what do you use to switch between buffers? C-x o seems really slow if i need to switch backwards


ido-switch-buffer seems nice (I use another thing myself)


there's also , never tried it


buffer != window


will ido-mode be in conflict with stuff like helm


@U45T93RA6 thanks, I was confused because Emacs' default binding for C-x o is other-window.


oh man i actually meant windows!


but i use C-o to create empty lines


C-o is nice, but lots of other emacs libs bind it, so I am forever having to unbind those... so it isn't ideal.


F keys maybe? (in which case, and if you use macOS, this is handy )


bind to shift and arrows


How does the memory usage of GraalVM compare to JVM's?


I found it about the same usage compared with OpenJDK


Philosophical CS debate question: How is all maths and arithmetic not just geometry? For any arithmetic construction, we're essentially building a machine, where spatial relations of the mechanism always matter ("no, carry the one to the left, not the right!"). And for that matter, how can one have an intuition about a scalar number without at least an intuition of a 1D line anyway?


is this a question about pedagogy or something about theory?


The spatial relations of the arithmetics may not exactly match 3D geometric intuition, but there's always some spatial relationship between each thing, perhaps along a 2D or 1D axis


Could be just pedagogical. Just a way of looking at it.


yeah. I think that it’s often useful to find different ways of explaining the same problem/solution


most people have some sort of spatial intuition, so explaining problems in a way that relates to that can really help


Yeah, my hunch though is that all algebraic intuitions are also geometric. Associativity, transitivity.. They're always constructed in relationship to order and placement of elements on a page. And the position of them matters.


mm, I think that is conflating “geometric” with the way we write


Well, even writing depends on the geometry of the textual elements


Like, associativity in algebra and reducing terms, carrying relations back and forth. We're operating over our idea of the arithmetic machine, and that things can be carried back and forth.


It's like a shorthand geometry


I think it’s useful to draw a distinction between geometry, the field and the representation of a problem


Yeah, they're sensibly distinct subjects in my mind too. Definitely a useful distinction


"carrying" is an artifact of a particular algorithm for addition. it is not innate to addition itself


Unless your "carrying" an apple to an apple cart 😉


> How is all maths and arithmetic not just geometry?


I'm just saying, all algorithms are carrying things in certain directions


that sounds pretty handwavy


all algorithms are hungry


for sufficiently useless and vague notions of carrying and hunger, respectively


True. And it's probably just a way of looking at it.


the greeks used to think that geometry was the fundamental expression of math


You could probably argue that all geometric relations can be represented in algebraic ones.


true but the converse doesn't hold

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But my suspicion is that we're all using geometric intuitions to construct those algebraic relations


See I doubt that


oh i doubt that very much


doing calculus recently with someone and it was entirely divorced from any intuition


it was symbolic manipulation 100%


these are the rules and you apply them


How is symbolic manipulation not a geometric activity?


the "carrying" algorithm for addition is symbolic manipulation. we aren't thinking of lines in a plane and how they combine. we are going over rules for symbols


Any symbolic manipulation is going to assume some geometric relationships between the symbols, afaict


that seems to me to mistake where the insight lies. just because it is written and physical doesn't mean that the insight into something is physical. the symbols d/dx(3x) = 3 doesn't use any geometric transformation. we aren't analyzing bezier curves. we could switch the symbols and it would not cloud our thinking. the insight is in the abstraction of functions and algebra, not in handwriting

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It may be one of those yin/yang things. Form and function. People always debating which side is more fundamental 🙂


From my perspective, any algorithm or equation you present, there will always be some spatially analogous transformation you're using to operate over the terms to arrive at some result


And some spatial transformations are analogous to others, so we end up with a bunch of spatial shorthand operations


what's an example of a spatial shorthand operation




IMO arithmetic is basically an automaton


We could add all the apples using ones, moving them around directly in space. But instead we put them in a line and give symbols to represent collective amounts of them, like 8 apples and 9 apples. Then we take those and put them in an automaton we call arithmetic, where if you move the things this way and that way, and carry the one, etc, you get the resulting number of apples you want. But that other transformation was still a spatial one, just over a more condensed analogous space. It was just a more general set of spatial relationships that we know are analogous, and we know one is more efficient than another.


Arithmetic is a 2D, 10 State automaton, with neighborhood rules specific to some columns and rows.


Well, 10 state in base 10 obviously


are you claiming this for all of math or just arithmetic


In base 2, it's basically a few basic xor type operations, going down the automaton


all maths


But, I'll grant you that the algebraic relationships may be just as... fundamental as the geometric ones.


this is russels paradox. Let R be the set of all sets which are not elements of themselves


if R is a member of itself, then by the definition, it is not a member of itself. If it is not a member of itself, then it meets the requirements of the rule that determines what is a member of itself


can you help me understand the geometric interpretation of this?


It's geometry that makes it a paradox, right?


can you point to the geometry for me?


Our intuitions of inclusion and exclusion


A circle that is both inside and outside of another circle


no circles involved


we can use heuristics and simple models to think of it but those are simply crutches. the idea is not based in geometry


But we can construct shapes in a certain number of dimensions that are paradoxical (incorrect) when translated into another set of dimensions


When a 2D picture is trying to represent a 3D shape, it can be paradoxically wrong about the feasibility of that transformation.


i have no idea what you're trying to say or how it relates to russel's paradox


again I think you’re conflating the way we can represent a problem with something fundamental about the problem


The semantics of the automaton in which Russel's expression was operating in are allowed to make expressions that fail to map to other geometric intuitions


just because we can represent a problem visually, doesn’t mean it has to do with geometry


I guess we're touching on the subject of whether truths exist as platonically disembodied objects, non-contingent on time and space. The whole Platonic vs Aristotelian interpretation


i don't think so. i thought you were saying you could essentially draw all of maths and it was a deep truth


Well, to the extent that a 3D geometric operation can be compiled down to a simplified 2D geometric operation over an automaton


As someone who has mostly studied math, I tend to agree with @dpsutton here. The progress we make when putting our reasoning in written form stems mostly from having a symbolic notation on a hard external medium - so it's symbolics more than it is geometry which are essential here. I could imagine a blind mathematician using touch, not sight, for this purpose. Granted, many mathematical intuitions are visual - but we tend to use writing (especially calculus) to make mechanical progress where our intuition cannot get us any further. What's beautiful with practicing maths IMHO is that it can expand your intuition to new territories, in particular ones which are not well captured visually.

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Quoting Von Neumann: "you don't understand math, you get use to it."


For the record I'm not trying to disparage one view point over another. Maybe I'm arguing that geometry is a little under appreciated, but I'd agree future progress is in understanding heavier maths. But I just can't help but see calculus as yet another automaton. Spatial operations that are conveniently more efficient than the spatial operations they analogize.


Sure, and if you can discover or promote ways in which geometry can fuel intuition I certainly won't deter you from that 🙂


... Yeah, I thought about it and I can't think of nothin' 🙂


It's mostly just a perspective on maths I guess


I always thought maybe some low hanging fruit was in those magic maths, where you get some arithmetic shortcuts for certain factors in certain bases.


And maybe you could exploit it using a fast base conversion scheme. Not sure if that comes from a geometric intuition though.


Most "real" maths tend to ignore residuals and fingerprints associated with clusters of number patterns in particular bases, right?


I guess it's slightly based on a geometric intuition: step outside the arithmetic axioms and exploit the invariant patterns that are incidental to the local neighborhood


i don't know what residuals and fingerprints are but they sound like numerical analysis might care about that kind of stuff. The thing i see from you is a lot of enthusiasm but just making these sweeping statements about invariant patterns and local neighborhoods. I don't know what you are trying to say. I think you should spend some time with some textbooks and refine what you want to say


Residues and fingerprints are in some textbooks I think


granted, IANAMathematician 😉


But it's not a bad idea. Basically, neighborhoods of numbers produced by arithmetic of a certain base will have information patterns in the numbers that are not exploited under the normal rules of arithmetic. You know the common ones: add up the digits of any 10 base number and if they equal 3, 6 or 9, then you know it's divisible by 3. Well, that's an artifact of base 10. If we all used base 50 instead, then we could add numbers together to see if 7 goes into them. Because 49 in that case is base-1.


But there's lots of other examples where different incidental relations between the numbers can be exploited


Sexagesimal arithmetic probably has a bunch of fast paths like that too


I'm not sure right now, but I think you can come up with a rule like that for numbers written in any base, and asking whether they are divisible by any integer N.


Some of them only need to look at the few least significant digits, others need to take all of the digits into account.


The bottleneck is the base conversion


But there's probably some magic math way to base convert by exponential steps. Then just stepwise to your goal.


and in a base notation that is not dependent on a limited number of unique digits


How about an arithmetic where every number had an imaginary/complex component as its base. So various numbers in different bases can coexists. Or even a single number can be represented internally by numbers of various bases, like 1,08(23b16),(86b23)59


So as to exploit some fast path between similar bases


So the above would be said, "one million, eighty twenty-three base sixteen thousand, eighty-six base twenty-three hundred and fifty nine" 🙂


You can of course represent numbers in a wide variety of ways other than the typical "base B" notation. You can think of hours, minutes, and seconds as a "variable base" notation if you squint hard enough, e.g. hours are base 12 or 24, minutes and seconds are base 60.


I'm sure there are some applications where alternate representations of numbers would give an advantage in computation speed, but I haven't worked with any of them enough to give examples.


Yeah, I've considered ((15)(19)) for number/base, so that you could even express a base as composite too ((15)((13)(16)))


Then do lambda recursion on base trees 🙂


But I wonder what that nested base lambda notation would look like with only binary numerals representing the numbers and their bases


I think I'd prefer the base to the left too, so it fits better with reduce and lambda semantics


Like, you could represent the number 168 in base 50 as ((50)(3)) ((50)(18)). Just add 3 and 18 and you get 21, which we already know is a multiple of 7 in base 50 (like 9 and 6 are to 3 in base 10)


And with respect to residues, for instance any 1 divided by 13 is 0.07692307692. That 076923 keeps repeating and when 13 interacts with other numbers, it tends to leave that fingerprint in lots of places. Some other primes also have their own "signatures" with respect to base ten. In base 13, obviously, that signature washes away. In other bases, it might also not show itself. And certain bases will show certain traces of certain primes left behind in their computations. Also an avenue of exploitation.


The signature of 7 in base ten is especially sticky in the sense that

user=> (* 2 142857)
user=> (* 3 142857)
user=> (* 4 142857)
user=> (* 5 142857)
user=> (* 6 142857)
It keeps rotating


Other numbers may have similar relationships to other bases that can be further exploited to detect common factors