Do or do not.... 
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- Edit 3Before modification by Joel at 11/03/2013 04:33:53 PM
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Kind of hard to say, there's a lot that goes through my head on reading 2X2 that doesn't seem parallel to normal written word. If math is a language, it is one that is sufficiently unlike others that I do not feel skill at it necessarily indicates skill at the others or vice versa. The correlation to me would be as awkward as assuming someone who was very good at identifying color and hue would have a noteworthy advantage at learning Russian. It literally seems as bizarre a connection to me as assuming skill at grammar implied better cooking skills form a heightened ability to interpret directions in a cookbook.
View original postIt is a logical process, just with different premises. I mean, you already do it to some extent: When you see "2X2" what comes to mind? I bet it is the words "two times two" not two pairs of objects. One COULD even say math is a special case of grammar, with its own supplementary set of operational rules, but that might be pushing things. Yes, I am a grammar pusher; the first verb's free....
Kind of hard to say, there's a lot that goes through my head on reading 2X2 that doesn't seem parallel to normal written word. If math is a language, it is one that is sufficiently unlike others that I do not feel skill at it necessarily indicates skill at the others or vice versa. The correlation to me would be as awkward as assuming someone who was very good at identifying color and hue would have a noteworthy advantage at learning Russian. It literally seems as bizarre a connection to me as assuming skill at grammar implied better cooking skills form a heightened ability to interpret directions in a cookbook.
The second example seems better to me. A big issue in teaching math, one you reference below, is that different people do conceptualize the operations and relations differently, so illustrations that convey them to some people are often useless to others.
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It's very hard to say, because we have so little casual common stuff and they had so little standardization. Anyone using an abacus is using positional notation and the concept is not tricky and virtually none of them had nearly all of their interactions of that sort with people taught the same as them. You and I see positional notation constantly and we think in it and everyone we know does too. However to humans about the only natural and shared view on math and counting is that we need grouping or tally to count anything above around 5 or 6. We can see five cars scattered randomly but near each other and just know 'five cars', anything much beyond that and we must count them or we need them grouped, we have to consciously think on it and we've been trained to a very common and standard way.
View original postFrom what I can tell though, positional notation was virtually unheard of in ancient numbers; they seem to have preferred additive notation, with the extra wrinkle of distinct symbol sets for larger and smaller numbers. One infamous example, courtesy Wikipedia, is that in Koiné (which it seems likely this die used) the number χξς would be understood as χ+ξ+ς, or 600+60+6. Interestingly, Wikipedia further notes that in his early revision of a Latin NT Jerome wrote that "The number 666 has been substituted for 616 either by analogy with 888, the [Greek] number of Jesus (Deissmann), or because it is a triangular number, the sum of the first 36 numbers (1+2+3+4+5+6...+36=666.)" Even then, it seems, the evils of science, or at least the related heresy, math, were corrupting the Holy Church. 
It's very hard to say, because we have so little casual common stuff and they had so little standardization. Anyone using an abacus is using positional notation and the concept is not tricky and virtually none of them had nearly all of their interactions of that sort with people taught the same as them. You and I see positional notation constantly and we think in it and everyone we know does too. However to humans about the only natural and shared view on math and counting is that we need grouping or tally to count anything above around 5 or 6. We can see five cars scattered randomly but near each other and just know 'five cars', anything much beyond that and we must count them or we need them grouped, we have to consciously think on it and we've been trained to a very common and standard way.
That is a very interesting phenomenon in itself, and one that has always fascinated me: That most people look at a randomly distributed group of up to five items and think "number," but look at anything else and think "lots." Most people can make it to six if presented that many objects arranged like the vertices of a hexagon, but must count the sides/corners of anything larger to know their number.
View original postI can attest to that since I was well into college before I ever learned 'long division' because the small gifted class I was in had the teacher opt to teach it when I'd go in for speech class (can't pronounce R's) because she showed me a couple of bigger division problems and I solved them on raw rapid multiplication skill and she assumed I knew how already, I was our star math student after all and the class was composed of kids who tended to have already learned a lot of stuff outside of school. Later in homeschooling it simply never came up and it wasn't till we were going over synthetic division that my math instructor realized I didn't know what the hell long division was. I was essentially doing it backwards and not as efficiently but retraining me to do it 'properly' was non-advantageous at that point, and so even though I know how to now I still use the old method. I learned formal geometry after learning trig and calc, and it alters my way of viewing geometric problems rather significantly, with advantages and disadvantages.
Interesting; makes me wonder what I was missing learning remedial handwriting while my classmates were, well, LEARNING.
' /> I have mixed feelings about formal geometry; exposure to formal logic was helpful, but it is definitely far more intuitive to me on a numerical basis.
View original postThis may factor into why I do not like the math/lang analogy because it rings false to me, I already know I start with the same representation and end with the same answer but follow a very different course to get there than most, and I keep my nose out for that in others and sometimes finding it, I tend to attribute some of my skill at instruction to that expectation and sensitivity to non-standard approaches by others. But in language this is totally different, I don't think two people read the same sentence, achieve an identical interpretation, yet process it differently. That just isn't how it works, they might get different interpretations of what was said but they don't start and end in the same place if they take different routes... I assume anyway
I am not so sure; different people tend to focus more, or first, on different parts of sentences, take a different perspective on each word/phrase/clauses significance, yet often arrive at the same destination. The real breakdown in the analogy is in your second point: Different people derive significantly different meanings from the same sentence far more often than from the same equation.
View original postIn a place where there was no formal education system for most and a very non-standard one for those who did get educated it would, IMO, be very probable for someone to casually invent positional notation in their own informal way and use it strictly for counting eggs by the dozen, teach it to their proximate colleagues an successors, and they all use it but never for anything else. Techno-speak is definitely not a modern invention.
Logical, though standardization does promote communication; a discoverys profundity does not matter much if the discoverer cannot "show their work" in a way that transmits it to others.
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I'd almost have to see it written out and annotated to grasp it. Remember that a lot of card players casually think in a parallel of base 13 superbase 4 but never view it that way and never apply it to anything but cards, even though they cheerfully make card analogies to life. If our clocks consisted of 4 periods, morning, afternoon, evening, and night, divided into 13 segments (27 minutes) and 52 'minutes' of 32 seconds subdivided into 52 'seconds' of .6 normal seconds you could be almost assured that card games and time would have all sort of common analogies and comparisons. "I'll meet you at club king for the film, I might be a suit late though" referring to a period of about half an hour and saying he might be abut 6 or seven minutes late. Or alternatively expressions like 'high noon' could work their way into cards. Any sort of competitive game or religious ritual are going to encourage those involved to rapidly assimilate the concept even if it has no outside parallel or logic and I think predispose them to try to graft that onto the outside world wherever there is any perceived overlap. Witness that 2d10 or d% is used to get a well known concept but a d20, with no daily use equivalent, generates them as 'natural 20!' or 'fuck, rolled a 1!' or even snake-eyes or boxcars. I don't think a game or religious divination would lead to adaptation for math or practical use but I could easily see existing math or common concept being brought into a game the way a d% is.
View original postRegardless, it LOOKS like they could have used multiple dice to generate larger numbers, and perhaps did, but the other dice would have used a completely different symbol set, additively. Another point of interest here is that the ancients generally mapped the digits 1-9 and their multiples of 10 and 100. That means there were 27 possible digits (forcing Koiné to modify three letters for use as numbers,) so a d20 could not contain all of them. Further, even if it held, say, the digits 1-10 and 10-100, it would have to omit any digits >10 that were not multiples of 10. To function as our d20 it would have to be enscribed with the digits 1-10, then the 10 digit followed by the digits 1-9 (or the reverse; by the Commutative Property of Addition it does not matter,) then the 20 digit.
I'd almost have to see it written out and annotated to grasp it. Remember that a lot of card players casually think in a parallel of base 13 superbase 4 but never view it that way and never apply it to anything but cards, even though they cheerfully make card analogies to life. If our clocks consisted of 4 periods, morning, afternoon, evening, and night, divided into 13 segments (27 minutes) and 52 'minutes' of 32 seconds subdivided into 52 'seconds' of .6 normal seconds you could be almost assured that card games and time would have all sort of common analogies and comparisons. "I'll meet you at club king for the film, I might be a suit late though" referring to a period of about half an hour and saying he might be abut 6 or seven minutes late. Or alternatively expressions like 'high noon' could work their way into cards. Any sort of competitive game or religious ritual are going to encourage those involved to rapidly assimilate the concept even if it has no outside parallel or logic and I think predispose them to try to graft that onto the outside world wherever there is any perceived overlap. Witness that 2d10 or d% is used to get a well known concept but a d20, with no daily use equivalent, generates them as 'natural 20!' or 'fuck, rolled a 1!' or even snake-eyes or boxcars. I don't think a game or religious divination would lead to adaptation for math or practical use but I could easily see existing math or common concept being brought into a game the way a d% is.
Though I feel there's something confused, rambling, and very much a massive digression to everything I wrote here 
Do not feel obliged to reply point for point
It is something of a sickness with me.
I will say THIS card player does NOT think in base 13 superbase 4, even when playing. Perhaps I SHOULD, but my memory is not in good enough shape to count every card; usually I just count honors so I know what is high in each suit, and distribution so I do not lead anything CERTAIN to be trumped (or worse, give the bad guys a rough-slough.) Sometimes that gets me in trouble once all the honors are gone and I cannot remember if an 8 or 7 or whatever is good or a higher non-honor spot is still out there. One such occasion proved especially embarrassing because I had lost count of the distribution as well, which left me wondering if my heart 8 (or 7, forget which) was high when it was not just the high heart, but the LAST heart.
From what I can tell, most people tend to think in terms of "un/somewhat/very likely," and do not go further absent the incentive you reference. In AD&D a natural 20 is a crit success and 1 is a crit fail (or vice versa,) while in GURPS a 3 or 4 is crit success and a 17 or 18 is a crit fail*. Most people will look at that and think "makes sense; criticals are supposed to be rare, or at least uncommon," some might even opine that the ability to produce either with two rolls rather than one makes them more common in GURPS. However, the chance of rolling 20 (or 1) on a d20 is a fairly respectable 5%, while the chance of rolling 17 or 18 on 3d6 is <2%—even though there are 4 times as many ways to do it! People who are not veteran gamers (or mathematicians) seldom realize that.
Anyway, to see it written out and annotated, try the below link.
*GURPS further complicates things because a natural 17 is only a "normal" failure for skills >15, and any natural roll 10+ below an unmodified skill is a crit success. Both incentivize buying skills past 15, which would otherwise be almost pointless since there are only 11/216 ways to roll >15 on 3d6.