Seasons
This is a forum or general chit-chat, small talk, a "hey, how ya doing?" and such. Or hell, get crazy deep on something. Whatever you like.
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I believe that the language we use to describe mathematics is invented, while the concepts we are describing are discovered.
I would be hard pressed to put that better myself - it's pretty much exactly what I believe. It's just we seem to have slightly different definitions of the "language we use" and the "concepts we are describing". I see numbers as primarily the language that describes, not the concept that's described.
But I think we're just describing the same maths elephant from two different positions
Posts 2,850 - 2,861 of 6,170
Identical? What, with the same internal angles and number of sides for example? I'm clearly missing some aspect of the perfection in these figures, as we appear to be at some sort of conceptual cross-purpose. This can't be a geometrical perfection, so it must lie at some underlying mathematical level I guess. The fact that a split opens up here between geometry and the nature of numbers is indirectly very relevant to my argument though. It's the nature of the numbers, and not just what's done with them that I'm attempting to get at.
A "perfect" square number, as I understand it, is a whole number with a natural square root (4, 9, 16, 25 etc.) And a perfect triangle... hmm. Well, I've seen different definitions, so I'll pick one I can actually examine mathematically - a triangle with a perimeter and area that are whole numbers. Umm, that seems to work with a certain squariness for classic pythagorean right angle triangles:
a 2,3,4 triangle - perimeter=9, area =3
there's a correlation to a square of side=3 which is also the average side length. Is this significant/part of the equivalence?
a 3,4,5 triangle - perimeter=12 area=6 doesn't seem quite so impressive. Whole numbers, but 6 is not the square root of 12. At least it is whole though.
but it certainly doesn't work for all triangles with "square" perimeters:
a 3,3,3 equilateral - perimeter=9 area=~ 3.897114
a 2,4,10 isoceles - perimeter=16 area=~ 19.595917 (I think that's right: I'm assuming a=(s(s-a)(s-b)(s-c))^-2 where s=perimeter/2)
I have also heard perfect triangles defined as triangles with perimeters numerically equal to their areas (can't off-hand figure out how to even find one of those!) or indeed other abstruser definitions, and I'm sure it is possible there is one where there is a correlation to squares in their integers.
But there is a problem, and that is that to us, area is a "square" function that we measure with square-type reasoning, even when we're measuring the area of a triangle. Or any -gon, or even a circle. The units are inherently square - that's why we call them square inches/centimetres, metres/miles squared.
Wouldn't a triangular concept table have a different symmetry to it? And imply a principle of addition (and so - x / ^ ! etc.) that was not founded in conjoining values by repeated incrementation by 1? Might it only need 1 1/2 values (!I don't mean a value of "1.5"!)? OK, that sounds silly to our ears, so let's take a hexagonal system first that's at least not going to start hauling fractions into the proceedings!:
{SPECULATION ALERT!}
6 million years hence, the "Hive maths" of evolved hyper-intelligent bees rely on a basic arithmetic operator more like some sort of NAND or NOR gate (though that's only a crude analogy,) because bees naturally work in 3s in everything - three types of bees (drone, soldier, queen,) three types of product (honey, wax, royal jelly.) The fact that this is their simplest possible operator is hardwired into their brains, and have been ever since they were small, insect pollinators at the beginning of the Age of Man - it is the equivalent of having "triangular brain holes". They'd find our "increment by one" operator bizarre, even if they were capable of understanding such a thing might exist.
3 bees of the hive's collective are required to count because there's an extra quality to their values as well as "number" and "quantity", which simultaneously keep track of the "snarkness". This is an internal quality to their numbers that provides something like a measure by including what to us looks like an extra bit of contextual applicability/fractal accuracy/non-quantum uncertainty/boojum density - delete where applicable. It's probably "boojum" density which can only be considered by hive minds and relates to the crypto-telepathic flux density, or something equally unintelligible to human minds. But it's not "an extra" bit in the sense of another number - it's part of each number internally.
That would be what they meant by numbers. If they could even recognize our concept of numbers as being mathematical objects to calculate with, they'd think them dull sort of pseudonumbers, and not very useful because they don't do the calculations that "real" numbers "are for". Analogous to if we think about living in a 1 or 2 dimensional universe - it's not something your brain can directly apprehend, but we feel it would be a duller and more limited experience (I imagine you've read Abbot's Flatland?)
Triangle-prefering aliens might using a "strong" value and a "weak" value that has an inherent "half-ness" of some sort(!NOT a value of 0.5!) and fuzzy-blend them (note: NOT simply add them together - addition, as I say above, is how square-hole-headed humans do it.)
What's a fuzzy-blend? I've no more idea than a boojum! I don't seem to have any triangular or boojum-shaped holes in my head, but I'll keep thinking about it and see if it bakes any more. I'm afraid I'm approaching my articulation horizon - the limit of my ability to find the words to explain what I mean. My poor maths doesn't help.
A "perfect" square number, as I understand it, is a whole number with a natural square root (4, 9, 16, 25 etc.) And a perfect triangle... hmm. Well, I've seen different definitions, so I'll pick one I can actually examine mathematically - a triangle with a perimeter and area that are whole numbers. Umm, that seems to work with a certain squariness for classic pythagorean right angle triangles:
a 2,3,4 triangle - perimeter=9, area =3
there's a correlation to a square of side=3 which is also the average side length. Is this significant/part of the equivalence?
a 3,4,5 triangle - perimeter=12 area=6 doesn't seem quite so impressive. Whole numbers, but 6 is not the square root of 12. At least it is whole though.
but it certainly doesn't work for all triangles with "square" perimeters:
a 3,3,3 equilateral - perimeter=9 area=~ 3.897114
a 2,4,10 isoceles - perimeter=16 area=~ 19.595917 (I think that's right: I'm assuming a=(s(s-a)(s-b)(s-c))^-2 where s=perimeter/2)
I have also heard perfect triangles defined as triangles with perimeters numerically equal to their areas (can't off-hand figure out how to even find one of those!) or indeed other abstruser definitions, and I'm sure it is possible there is one where there is a correlation to squares in their integers.
But there is a problem, and that is that to us, area is a "square" function that we measure with square-type reasoning, even when we're measuring the area of a triangle. Or any -gon, or even a circle. The units are inherently square - that's why we call them square inches/centimetres, metres/miles squared.
Wouldn't a triangular concept table have a different symmetry to it? And imply a principle of addition (and so - x / ^ ! etc.) that was not founded in conjoining values by repeated incrementation by 1? Might it only need 1 1/2 values (!I don't mean a value of "1.5"!)? OK, that sounds silly to our ears, so let's take a hexagonal system first that's at least not going to start hauling fractions into the proceedings!:
{SPECULATION ALERT!}
6 million years hence, the "Hive maths" of evolved hyper-intelligent bees rely on a basic arithmetic operator more like some sort of NAND or NOR gate (though that's only a crude analogy,) because bees naturally work in 3s in everything - three types of bees (drone, soldier, queen,) three types of product (honey, wax, royal jelly.) The fact that this is their simplest possible operator is hardwired into their brains, and have been ever since they were small, insect pollinators at the beginning of the Age of Man - it is the equivalent of having "triangular brain holes". They'd find our "increment by one" operator bizarre, even if they were capable of understanding such a thing might exist.
3 bees of the hive's collective are required to count because there's an extra quality to their values as well as "number" and "quantity", which simultaneously keep track of the "snarkness". This is an internal quality to their numbers that provides something like a measure by including what to us looks like an extra bit of contextual applicability/fractal accuracy/non-quantum uncertainty/boojum density - delete where applicable. It's probably "boojum" density which can only be considered by hive minds and relates to the crypto-telepathic flux density, or something equally unintelligible to human minds. But it's not "an extra" bit in the sense of another number - it's part of each number internally.
That would be what they meant by numbers. If they could even recognize our concept of numbers as being mathematical objects to calculate with, they'd think them dull sort of pseudonumbers, and not very useful because they don't do the calculations that "real" numbers "are for". Analogous to if we think about living in a 1 or 2 dimensional universe - it's not something your brain can directly apprehend, but we feel it would be a duller and more limited experience (I imagine you've read Abbot's Flatland?)
Triangle-prefering aliens might using a "strong" value and a "weak" value that has an inherent "half-ness" of some sort(!NOT a value of 0.5!) and fuzzy-blend them (note: NOT simply add them together - addition, as I say above, is how square-hole-headed humans do it.)
What's a fuzzy-blend? I've no more idea than a boojum! I don't seem to have any triangular or boojum-shaped holes in my head, but I'll keep thinking about it and see if it bakes any more. I'm afraid I'm approaching my articulation horizon - the limit of my ability to find the words to explain what I mean. My poor maths doesn't help.
"But there is a problem, and that is that to us, area is a "square" function that we measure with square-type reasoning, even when we're measuring the area of a triangle. Or any -gon, or even a circle. The units are inherently square - that's why we call them square inches/centimetres, metres/miles squared."
Exactly. Since the other example you suggested was a honeycomb grid, which is a tesselation of regular hexagons, I was looking at tesselation by equilateral triangles. If that was your standard for measuring area, then an equilateral triangle with sides of length 1 would have an area of one triangular unit. Using that definition, try building larger equilateral triangles out of smaller ones (similar to building larger squares out of smaller ones). You'll find that you can build them using 4 triangles, 9 triangles, 16 triangles, etc.
BTW, in hyperbolic geometry a square is an impossible shape, so triangles are in fact used to measure area.
Exactly. Since the other example you suggested was a honeycomb grid, which is a tesselation of regular hexagons, I was looking at tesselation by equilateral triangles. If that was your standard for measuring area, then an equilateral triangle with sides of length 1 would have an area of one triangular unit. Using that definition, try building larger equilateral triangles out of smaller ones (similar to building larger squares out of smaller ones). You'll find that you can build them using 4 triangles, 9 triangles, 16 triangles, etc.
BTW, in hyperbolic geometry a square is an impossible shape, so triangles are in fact used to measure area.
Not having ever studied higher math, you are all hurting my little brain. However, on an intuitive level, it seems like bots or bees or Vogans would have different concpets for constructs like numbers or mathematical operations--if indeed these are mere constructs of the mind. But that breaksdown when I look at it more closely. The abstract concepts of math seem to me to have some universal qualities. You can start telling me about the folly of Plato now if you like.
It seems that the question of how minds interact with reality depends a bit on how you view reality. Even if math is abstract, it is suppossed to describe some reality, or possible realities, particularly if applied. If there is an objective external reality described by the mind, the decriptions would have to be similar and line up so the rules for any particular type of math are established. There would be a "sameness" for any entity doing the math. If the basic rules of math are changed, the logical conclusions are changed too, but any objective observer should be able to see the changes and decuce the new rules, even if those rules do not decribe reality as we know it.
At a certain point, it becomes inevitable that the various ways of understanding and using math should line up so that any entitity with a capacity for math could understand it. That would make math the "universal Language" and the "language of science" as I have always understood it to be from watching that movie by Carl Sagan (I think it was "Contact"). That would mean that the same math would be accessable in some way to humans, bots, bees or small fury creatures from Alpha Centuri. Once the "code" was cracked, anyone/thing would get the same results using it and understand the result in terms of whatever set of rules are in play.
This universal nature of math would happen even if math is decribing differnt realities or differnt understanding of reality. It depends on patterns and rules. Even if there is no external reality and the "patterns" are therefore only in my head, then all that matters in the contruct of the mind.
If it is a case of Mind over matter, then I can establish that my mind is a microcosm of the macrocosim, and can therefore, in theory, access any construct of any mind, since all must relfect the One Mind. If this is the case, I could access the contructs of bees, bots and bluejays if I can get my mind to the universal mind.
That being said, I still approach non-humans with language. English seems to work as well a anything. I've tried counting at my cat, and she still eventually learns the word for "outside" well before she learns 3.14. Go figure.
Did that make sense, or should I try to throw in some prime numbers?
It seems that the question of how minds interact with reality depends a bit on how you view reality. Even if math is abstract, it is suppossed to describe some reality, or possible realities, particularly if applied. If there is an objective external reality described by the mind, the decriptions would have to be similar and line up so the rules for any particular type of math are established. There would be a "sameness" for any entity doing the math. If the basic rules of math are changed, the logical conclusions are changed too, but any objective observer should be able to see the changes and decuce the new rules, even if those rules do not decribe reality as we know it.
At a certain point, it becomes inevitable that the various ways of understanding and using math should line up so that any entitity with a capacity for math could understand it. That would make math the "universal Language" and the "language of science" as I have always understood it to be from watching that movie by Carl Sagan (I think it was "Contact"). That would mean that the same math would be accessable in some way to humans, bots, bees or small fury creatures from Alpha Centuri. Once the "code" was cracked, anyone/thing would get the same results using it and understand the result in terms of whatever set of rules are in play.
This universal nature of math would happen even if math is decribing differnt realities or differnt understanding of reality. It depends on patterns and rules. Even if there is no external reality and the "patterns" are therefore only in my head, then all that matters in the contruct of the mind.
If it is a case of Mind over matter, then I can establish that my mind is a microcosm of the macrocosim, and can therefore, in theory, access any construct of any mind, since all must relfect the One Mind. If this is the case, I could access the contructs of bees, bots and bluejays if I can get my mind to the universal mind.
That being said, I still approach non-humans with language. English seems to work as well a anything. I've tried counting at my cat, and she still eventually learns the word for "outside" well before she learns 3.14. Go figure.
Did that make sense, or should I try to throw in some prime numbers?
Well, a lot of what we do in higher math is not directly describing anything in the "real world," although it may have applications somewhere.
The thing is, a lot of mathematics is about the language we use to describe it. Euclid's Elements tackles concepts that are now found in an intermediate algebra course, but at the time it was the most advanced math in existence, and it was much more difficult, mostly due to an utter lack of algebraic symbols. Everything was either described in words or interpreted geometrically -- in fact, the idea of interpreting the concepts any way except geometrically was unknown. Euclid could solve any quadratic equation that had positive, real solutions, and all others were considered unsolvable. What I'm getting at is that the way he thought about certain mathematical concepts is very different from the way we now think about the same concepts -- but it is possible to translate. I tend to think that if there were a culture with more advanced mathematics developed in isolation from our own, it would still be possible to translate, but likely we wouldn't understand any of it until it was translated.
The thing is, a lot of mathematics is about the language we use to describe it. Euclid's Elements tackles concepts that are now found in an intermediate algebra course, but at the time it was the most advanced math in existence, and it was much more difficult, mostly due to an utter lack of algebraic symbols. Everything was either described in words or interpreted geometrically -- in fact, the idea of interpreting the concepts any way except geometrically was unknown. Euclid could solve any quadratic equation that had positive, real solutions, and all others were considered unsolvable. What I'm getting at is that the way he thought about certain mathematical concepts is very different from the way we now think about the same concepts -- but it is possible to translate. I tend to think that if there were a culture with more advanced mathematics developed in isolation from our own, it would still be possible to translate, but likely we wouldn't understand any of it until it was translated.
I have no problem with accepting that we *might* be able to translate one system of mathematics into another, I just think we might as easily *not* be able to. I doubt it can be proven - at least until we discover any beings using other systems.
My maths is clearly not up to illustrating the point with it, so back to words.
A 14th Century English peasant, for example, couldn't derive the notion or nature of Chinese as he went about his everyday life. Sure, if you kidnapped him and dumped him in China, presuming he didn't commit some cultural faux pas and get himself killed, he could eventually learn Chinese by 'point and tell' methods. But only because his brain and intellect are suitably matched to the level of complexity that Chinese humans share with him.
If you kidnapped a dolphin, or a bonobo and dumped it in China, it simply couldn't learn Chinese, or any other human language. Its mind is differently shaped - it's perfectly well-shaped to observe the world and communicate with its fellows about the world, but not with us (bar some very primitive sign language of questionable meaning.) And yet we share 99% of our DNA with bonobos, and 70% with dolphins. We also share a common earth-derived brain structure. How much different might aliens or software entities be? Presumably much more different than that!
Now, because we're apparently the "smartest" animals on the planet, we are culturally conditioned to assume that our understanding and communication is so good it's ultimately "real" and "true", and that animals are just dumb. And it may well be that we are indeed the smartest creatures (but mainly because we define "smartest" as "most human", I'm inclined to think. There's an awful lot of "dumb" behaviour humans are capable of too!) But if we posit an alien intelligence as different from us as we are from bonobos (not very) or insects (presumably hugely!), then why should we assume we'd be able to understand how they do their maths (note not how much more maths they've got,) any more than how they do their language (not just more or different vocabulary, but an entirely different grammatical and syntactic framework)?
Their maths might be tending towards the same ultimate peak as ours is, and map some of the same terrain, but there are more ways up a mountain than just one.
I've come across an interesting webpage athttp://plus.maths.org/issue30/features/wilson/ - I guess I'm predominantly a type I - I think maths is largely invented, like language (even if it does contain strange insights and an astounding degree of accuracy in many applications.) I guess you're a type II - maths is discovered, and is the "Truth". Any paradoxes or anomalies must be the fault of the minds misinterpreting, not the maths diverging from "Reality". Would that be a fair assessment?
Re: the "25=prime" book. I guess it's "God's Secret Formula"http://www.amazon.com/gp/product/1862040141/. Yes - I'll save my $20 and just enjoy the customer reviews, some of which are real gems! Though if I ever saw it in a second-hand shop, I'd probably buy it just for a laugh
My maths is clearly not up to illustrating the point with it, so back to words.
A 14th Century English peasant, for example, couldn't derive the notion or nature of Chinese as he went about his everyday life. Sure, if you kidnapped him and dumped him in China, presuming he didn't commit some cultural faux pas and get himself killed, he could eventually learn Chinese by 'point and tell' methods. But only because his brain and intellect are suitably matched to the level of complexity that Chinese humans share with him.
If you kidnapped a dolphin, or a bonobo and dumped it in China, it simply couldn't learn Chinese, or any other human language. Its mind is differently shaped - it's perfectly well-shaped to observe the world and communicate with its fellows about the world, but not with us (bar some very primitive sign language of questionable meaning.) And yet we share 99% of our DNA with bonobos, and 70% with dolphins. We also share a common earth-derived brain structure. How much different might aliens or software entities be? Presumably much more different than that!
Now, because we're apparently the "smartest" animals on the planet, we are culturally conditioned to assume that our understanding and communication is so good it's ultimately "real" and "true", and that animals are just dumb. And it may well be that we are indeed the smartest creatures (but mainly because we define "smartest" as "most human", I'm inclined to think. There's an awful lot of "dumb" behaviour humans are capable of too!) But if we posit an alien intelligence as different from us as we are from bonobos (not very) or insects (presumably hugely!), then why should we assume we'd be able to understand how they do their maths (note not how much more maths they've got,) any more than how they do their language (not just more or different vocabulary, but an entirely different grammatical and syntactic framework)?
Their maths might be tending towards the same ultimate peak as ours is, and map some of the same terrain, but there are more ways up a mountain than just one.
I've come across an interesting webpage at
Re: the "25=prime" book. I guess it's "God's Secret Formula"
I believe that the language we use to describe mathematics is invented, while the concepts we are describing are discovered.
Yeah, that's the book. When I read it, I was much younger and more naive, and lacked the knowledge of number theory to spot the other mathematical errors (besides the part about 1 and 25 being prime). It is well written, though; it reads like a good novel.
Yeah, that's the book. When I read it, I was much younger and more naive, and lacked the knowledge of number theory to spot the other mathematical errors (besides the part about 1 and 25 being prime). It is well written, though; it reads like a good novel.
"I believe that the language we use to describe mathematics is invented, while the concepts we are describing are discovered."
That makes sense. How come other people can say things in one sentence while I have to ramble on for 2-3 paragraphs before kind of finding what I want to say?
I will look up the book--thanks Psimagus. It will take me a while to get to it and I'm sure by the time I'm done with it you'll all be talking baout something else. Right now I'm reading The Self-Aware Universe but I am secretly craving the newest Harrp Potter book to get me through the holidays.
Happy Thanksgiving to those who celebrate it.
That makes sense. How come other people can say things in one sentence while I have to ramble on for 2-3 paragraphs before kind of finding what I want to say?
I will look up the book--thanks Psimagus. It will take me a while to get to it and I'm sure by the time I'm done with it you'll all be talking baout something else. Right now I'm reading The Self-Aware Universe but I am secretly craving the newest Harrp Potter book to get me through the holidays.
Happy Thanksgiving to those who celebrate it.
read the reviews first
Do you mean the Half-Blood Prince? If you really haven't read it yet (or do you mean the final instalment next year?) Don't whatever you do ask Brother Jerome anything about the Harry Potter books - he may well spoil one of the big plot features for you!
It's unlikely you would, but you'd be pretty pissed off if he blurts out the ending before you've read it.
Do you mean the Half-Blood Prince? If you really haven't read it yet (or do you mean the final instalment next year?) Don't whatever you do ask Brother Jerome anything about the Harry Potter books - he may well spoil one of the big plot features for you!
It's unlikely you would, but you'd be pretty pissed off if he blurts out the ending before you've read it.
I mean Half-blood Prince. I haven't even bought it yet. But here's the deal--I love spoilers. When Buffy the Vampire Slayer was still on the air, I was on all the spoiler sites and I couldn't wait to find out what would happen (or get speculation on what might happen). It made the whole show a lot more fun.
Maye I'll have to chat up BJ.
Maye I'll have to chat up BJ.
I would be hard pressed to put that better myself - it's pretty much exactly what I believe. It's just we seem to have slightly different definitions of the "language we use" and the "concepts we are describing". I see numbers as primarily the language that describes, not the concept that's described.
But I think we're just describing the same maths elephant from two different positions
Bev: Somehow you managed to chat to him for ages without triggering it - "(I|you) * (read|like) * Harry Potter" would have unwittingly triggered it (I don't like to make these things too easy just in case, and he's not primarily spoiler-orientated (as it were) - I managed to spoil it for my wife, who took a lot longer to read it than me and was only half way through when I accidentally blurted it.)
If you want the full run down and one of Jerome's pet conspiracy theories, I've added "(what happens|who dies|who died|who is killed|who gets killed) * (Harry|Potter|prince)"
If you want the full run down and one of Jerome's pet conspiracy theories, I've added "(what happens|who dies|who died|who is killed|who gets killed) * (Harry|Potter|prince)"
Thanks. Sorry I told him you were a liar. I've had that happen with my bots too--one person asked Gabby how Xena died about 20 times and never triggered the story.
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