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.
Posts 2,838 - 2,849 of 6,170
Eugene:
Why do you think that an electronic lifeform wouldn't view whole numbers as special? There's an obvious distiction between rational and irrational numbers, and rationals are derived from whole numbers.
I don't say they wouldn't, just that they wouldn't
necessarily. In fact I think they probably will, since they will be in many senses, a product of human mathematics.
There's an "obvious" distinction? Yes. It is indeed obvious - that's precisely what it is. It's just as obvious as the idea that the sun and moon and stars go round the earth, or that heavy objects fall faster than light ones. Or Dubya's conviction that "obviously" we must attack Iraq, Or Osama's conviction that "obviously" all imperialist infidels must die. "Obvious" is a function of our brains, not of the "World that Is".
So where are these numbers that are so obviously distinct? Point to one for me.
Am I saying we should give up mathematics? Absolutely no! It's the finest symbological system we have yet had to interpret much of what we perceive around us, and is continuing to prove itself very useful.
Another top level symbological system is language, and we shouldn't give up on that one either. Symbology is the only way our brains work.
We can point around and say that is a stick, that is another stick. There are two sticks. Obviously the first one I point to is number one, and the second one I point to is number two. But if I burn the first stick in my campfire, where is the second stick's "secondness" now?
Where is the "two-ness" it had/was?
Was it in the spatial position of the stick? No. I haven't touched it, but the "two-ness" of the second one has still disappeared. To been replaced with a "one-ness", if we choose to count again.
Was it inside the stick as some sort of quality it possessed? No, or it would surely still possess it. We only destroyed the other stick.
Was it in the relative distance between the sticks? No. We can move them around instead of burning one, but if we still count them both, they're still one and two.
Was it in the "mind" of the stick? No, AFAWK sticks don't possess conscious minds.
So it was... in our minds all along. Just an interpretation of a relationship that we imposed on the sticks in the first place.
Ulrike:
I'm disappointed in you
The Tao is one, indivisible, all-encompassing... What exactly is that but a meta-rule to identify divergence between Reality and human symbological interpretations of it?
True, and it's a trivial cheat on my part to translate the number not the quantity and silently transform prime 17(base10) into 25(base6) and then substitute it for the quoted value, since there are still this many "x"s in the original count, whichever base we use: xxxxxxxxxxxxxxxxxxxxxxxxx. And that's 41(base6)=25(base10), and not prime according to our accepted definition. At least I'm assuming the book didn't specify base 6, or anything as trivially resolvable as such partial transformation for what on the face of it is "obviously wrong". (Of course, "assuming" things is one of our greatest strengths and one of our greatest weaknesses.)
It's rather curious that 41(base10) is also prime do you think?
prime 17(base10)=25(base6)
non-prime 25(base10)=41(base6)
prime 41(base10)=105(base6)
Is that sort of behaviour common (-er than random distribution would suggest,) with alternate translation of number and quantity do you know? it's a shame neither 105 or 253 are prime, but it would be interesting to see the distribution.
But the explanation of how primes are factored (which is of course totally correct in mathematics as we use it,) surely assumes that the grid is of squares.
Might bots or aliens not prefer triangular/pyramidal roots, and determine primes by reference to triangular grids? Intelligent bees might even use hexagonal roots if Man were to die out and bees considerably evolve to become the dominant intelligence on Earth.
We choose squares because our minds find them easier to manipulate than other shapes. As if we had square holes in our minds that best suit square pegs. But if an alien had triangular holes, or bees had hexagonal holes, square tables would look ridiculous, and give all the wrong answers. Of course, that has some pretty heavy-duty implications as to the nature of potential non-human numbers in themselves, and not just in how they're used!
Not strictly relevant to the above, but an indication of why we should perhaps be wary of the "obvious" perhaps:
When the wheel was first invented it took a few thousand years to iron out the bugs. The wheels made by Seller-of-Used-Rocks were square, and all of his customers complained about the bumpy ride. One day his best customer, Basher-of-Small-Furry-Animals, was visiting his workshop. Seller-of-Used-Rocks proudly displayed his newest invention. "Here, Basher, is my new improved wheel. I call it the 'rolleasy'. Isn't she a beauty? Ten skins I'm going to charge; but to you, my friend,a real bargain at six skins each, provided you buy two pairs and a spare."
Basher-of-Small-Furry-Animals stared at the new wheel with some puzzlement. Eventually he said, "But it's triangular!"
"Of course," replied Seller-of-Used-Rocks.
"How can that be an improvement?"
"Don't you see? It's obvious!" enthused Seller-of-Used-Rocks. "One less bump."
Posts 2,838 - 2,849 of 6,170
Why do you think that an electronic lifeform wouldn't view whole numbers as special? There's an obvious distiction between rational and irrational numbers, and rationals are derived from whole numbers.
I don't say they wouldn't, just that they wouldn't
necessarily. In fact I think they probably will, since they will be in many senses, a product of human mathematics.
There's an "obvious" distinction? Yes. It is indeed obvious - that's precisely what it is. It's just as obvious as the idea that the sun and moon and stars go round the earth, or that heavy objects fall faster than light ones. Or Dubya's conviction that "obviously" we must attack Iraq, Or Osama's conviction that "obviously" all imperialist infidels must die. "Obvious" is a function of our brains, not of the "World that Is".
So where are these numbers that are so obviously distinct? Point to one for me.
Am I saying we should give up mathematics? Absolutely no! It's the finest symbological system we have yet had to interpret much of what we perceive around us, and is continuing to prove itself very useful.
Another top level symbological system is language, and we shouldn't give up on that one either. Symbology is the only way our brains work.
We can point around and say that is a stick, that is another stick. There are two sticks. Obviously the first one I point to is number one, and the second one I point to is number two. But if I burn the first stick in my campfire, where is the second stick's "secondness" now?
Where is the "two-ness" it had/was?
Was it in the spatial position of the stick? No. I haven't touched it, but the "two-ness" of the second one has still disappeared. To been replaced with a "one-ness", if we choose to count again.
Was it inside the stick as some sort of quality it possessed? No, or it would surely still possess it. We only destroyed the other stick.
Was it in the relative distance between the sticks? No. We can move them around instead of burning one, but if we still count them both, they're still one and two.
Was it in the "mind" of the stick? No, AFAWK sticks don't possess conscious minds.
So it was... in our minds all along. Just an interpretation of a relationship that we imposed on the sticks in the first place.
I'm disappointed in you
The Tao is one, indivisible, all-encompassing... What exactly is that but a meta-rule to identify divergence between Reality and human symbological interpretations of it?
no. Of course not. Mathematics is the best system we have for certain classes of problem solving, and this includes many involving observations of what we perceive to be a "mechanical nature" of phenomena in the universe around us.
But just because it explains many complex effects using simple rules doesn't mean the rules are "right", or meaningful in any other context than the wiring of our brains provides. You'll be telling me next you think a cannonball dropped off a tower "follows the law of gravity" next!
I tend to prefer logic, probably because I've studied more logic than mathematics - as Keynes said, "first logic, then mathematics if appropriate".But I don't seek to defend that on any absolute grounds. It's just a preference I share with Keynes, and I also say that logic is as much a construct of human symbology as mathematics. How accurately either of them map any underlying reality is unquantifiable to or by us, and probably ultimately meaningless.
But just because it explains many complex effects using simple rules doesn't mean the rules are "right", or meaningful in any other context than the wiring of our brains provides. You'll be telling me next you think a cannonball dropped off a tower "follows the law of gravity" next!
I tend to prefer logic, probably because I've studied more logic than mathematics - as Keynes said, "first logic, then mathematics if appropriate".But I don't seek to defend that on any absolute grounds. It's just a preference I share with Keynes, and I also say that logic is as much a construct of human symbology as mathematics. How accurately either of them map any underlying reality is unquantifiable to or by us, and probably ultimately meaningless.
I should probably have expanded slightly on my "Of course not", since on reflection it won't be clear to anyone but myself.
The problems that various formalist (or even deductivist) schema fall into seem to me just as serious as the problems that afflict realists. The problematic epistemology of assigning meaning to the symbols you're using are unresolved, and Goedel has demonstrated (to my own satisfaction anyway,) that any possible logico-mathematical system can never resolve this to prove its own consistency.
I would probably describe myself as vaguely some sort of logicist with intuitionist leanings (if I really have to adopt yet another label - I am of course not a "logicist", I am me.) Or adopt an entirely new label - I am a "cognitist", if you like. But I am saddled by a poor understanding of many branches of mathematics, and a personal preference for semantics.
The problem is that mathematics is wonderful for telling us what happens. But useless for telling us why or how it happens. A cannonball dropped off a tower doesn't follow the law of gravity - there is no mechanism in it that calculates the gravitational field its in, and the friction of the air through which its passing and then determines the appropriate acceleration. It would be more accurate (but still, I think missing the point,) to say that the law of gravity follows the cannonball. The falling of the ball and its accurate match to the motion derived from the "law" that humans have devised is certainly impressive, but I think these phenomena are ultimately contingent, not causal. Indeed, causality itself appears to me to be a strong candidate for regarding as an entirely "unreal" human construct.
I am guessing you would consider yourself to be some sort of "constructionist"? Or are you an unregenrate Plato-realist?
The problems that various formalist (or even deductivist) schema fall into seem to me just as serious as the problems that afflict realists. The problematic epistemology of assigning meaning to the symbols you're using are unresolved, and Goedel has demonstrated (to my own satisfaction anyway,) that any possible logico-mathematical system can never resolve this to prove its own consistency.
I would probably describe myself as vaguely some sort of logicist with intuitionist leanings (if I really have to adopt yet another label - I am of course not a "logicist", I am me.) Or adopt an entirely new label - I am a "cognitist", if you like. But I am saddled by a poor understanding of many branches of mathematics, and a personal preference for semantics.
The problem is that mathematics is wonderful for telling us what happens. But useless for telling us why or how it happens. A cannonball dropped off a tower doesn't follow the law of gravity - there is no mechanism in it that calculates the gravitational field its in, and the friction of the air through which its passing and then determines the appropriate acceleration. It would be more accurate (but still, I think missing the point,) to say that the law of gravity follows the cannonball. The falling of the ball and its accurate match to the motion derived from the "law" that humans have devised is certainly impressive, but I think these phenomena are ultimately contingent, not causal. Indeed, causality itself appears to me to be a strong candidate for regarding as an entirely "unreal" human construct.
I am guessing you would consider yourself to be some sort of "constructionist"? Or are you an unregenrate Plato-realist?
These forums are the one place in where I DON'T feel like the most/only intelligent person in the room.
Reading things you've written is a humbling experience, psimagus...
Reading things you've written is a humbling experience, psimagus...
Oh for goodness' sake don't go getting humble - it's not like I have the answers. I just like the questions
I don't know what I am in that regard. I find the formalist argument hard to swallow, considering the intimate connection of mathematics with science, but have yet to see or come up with a good counter-proposal. The question, I think, boils down to the relationship between "number" and "quantity". I didn't come up with that idea, BTW -- it was in a book by a guy with some very strange ideas about the universe. He intrigued me right up to the point where he said that 25 was prime.
What do you mean by "contingent" rather than "causal"?
What do you mean by "contingent" rather than "causal"?
Indeed, formalism/realism/constructionism/intuitionism/... they strike me, like so many
systems of systems as probably all being right in limited ways.
Take the five blind men who examine an elephant: one feels the trunk and says "it's like a
great big snake."
One finds a leg and says "no, it's like a big warm tree trunk. And hey, it moves too! A
large warm, walking tree!"
One feels an ear and says "No it's not - it's like a horrid, flappy bat with leathery
wings."
One blunders underneath it and says "No it's primarily an opening, like a door-frame, but it's not tall enough to walk through without stooping, and the lintel is saggy and soft so you don't damage your head when you bang into it. That's a useful design feature, but would have been better if it was taller and had a door mounted in it to keep the wind out."
The last blind man walks round the back and is hit on the head by a large quantity of dung:
"Uuurrgghh! I don't know what it's like. But I don't like it!"
Which one is right? They all are in a limited sense.
I'm inclined to agree with the last one most though - if we're blind we can't know it
absolutely, yet we can still pass valid opinions on it. If we wave a magic wand and give them sight - just one more possible sense to intepret the world with, they are just like us. But still limited to five senses and brains of similar construction and size. We still can't know the elephant absolutely. But we can still observe it and express opinions.
That sounds like an interesting book. Do you recall the title/author. I wouldn't mind
reading that. 25 is of course prime in base 6 (hmm, or is it...? Are primes a mathematical
or numerical phenomenon - oh I don't know, it's too early in the day for me!) but I don't
suppose he meant it that way
Indeed "number and quantity" - that's precisely the nub of the problem. And as a religious mathematician, do you not consider that God need not be bound by inevitable correlation of the two? There is the Trinity, for example to demonstrate that the proposition 1=3 need not be meaningless (I shan't say absurd).
"Contingent", Yes, that was a best fit word that I'm still not wholly happy with. There are
several definitions, some having to do with causality which is specifically NOT
what I intend (which is evident from the context I think.) But the residual meaning of the
latin contingere>contingerus is also acceptable in English, and is the one I intend.
Happening together, linked in some subtler way than by mere causality. It is the nature of that link that fascinates me.
I considered "congruent" (too misleadingly redefined over the years into geometric terms,)
and "coincident". Coincidere>coinciderus would fit it very well, but in English the
term has picked up so many connotations of chance/non-inevitability that it would have been
perverse and misleading to use it.
But there you go - words are just symbols, and don't perfectly map the underlying Reality. If they did, they would BE the underlying Reality. I can't see numbers are qualitatively different in themselves.
systems of systems as probably all being right in limited ways.
Take the five blind men who examine an elephant: one feels the trunk and says "it's like a
great big snake."
One finds a leg and says "no, it's like a big warm tree trunk. And hey, it moves too! A
large warm, walking tree!"
One feels an ear and says "No it's not - it's like a horrid, flappy bat with leathery
wings."
One blunders underneath it and says "No it's primarily an opening, like a door-frame, but it's not tall enough to walk through without stooping, and the lintel is saggy and soft so you don't damage your head when you bang into it. That's a useful design feature, but would have been better if it was taller and had a door mounted in it to keep the wind out."
The last blind man walks round the back and is hit on the head by a large quantity of dung:
"Uuurrgghh! I don't know what it's like. But I don't like it!"
Which one is right? They all are in a limited sense.
I'm inclined to agree with the last one most though - if we're blind we can't know it
absolutely, yet we can still pass valid opinions on it. If we wave a magic wand and give them sight - just one more possible sense to intepret the world with, they are just like us. But still limited to five senses and brains of similar construction and size. We still can't know the elephant absolutely. But we can still observe it and express opinions.
That sounds like an interesting book. Do you recall the title/author. I wouldn't mind
reading that. 25 is of course prime in base 6 (hmm, or is it...? Are primes a mathematical
or numerical phenomenon - oh I don't know, it's too early in the day for me!) but I don't
suppose he meant it that way
Indeed "number and quantity" - that's precisely the nub of the problem. And as a religious mathematician, do you not consider that God need not be bound by inevitable correlation of the two? There is the Trinity, for example to demonstrate that the proposition 1=3 need not be meaningless (I shan't say absurd).
"Contingent", Yes, that was a best fit word that I'm still not wholly happy with. There are
several definitions, some having to do with causality which is specifically NOT
what I intend (which is evident from the context I think.) But the residual meaning of the
latin contingere>contingerus is also acceptable in English, and is the one I intend.
Happening together, linked in some subtler way than by mere causality. It is the nature of that link that fascinates me.
I considered "congruent" (too misleadingly redefined over the years into geometric terms,)
and "coincident". Coincidere>coinciderus would fit it very well, but in English the
term has picked up so many connotations of chance/non-inevitability that it would have been
perverse and misleading to use it.
But there you go - words are just symbols, and don't perfectly map the underlying Reality. If they did, they would BE the underlying Reality. I can't see numbers are qualitatively different in themselves.
In base 6, 11 = 5*5. In base 2, 11001 = 101*101. Prime factorization is a mathematical property, not dependent on the base. A grid of squares with five squares to a side contains the same number of squares no matter how you describe them.
Ummm... I think you mean 41 = 5*5 in base 6. Note that in base 10, 41 is prime, but is describing an entirely different quantity. A quantity stays the same however we label it, and if that quantity can be arranged in a rectangle other than 1xQuantity, then it's not prime.
True, and it's a trivial cheat on my part to translate the number not the quantity and silently transform prime 17(base10) into 25(base6) and then substitute it for the quoted value, since there are still this many "x"s in the original count, whichever base we use: xxxxxxxxxxxxxxxxxxxxxxxxx. And that's 41(base6)=25(base10), and not prime according to our accepted definition. At least I'm assuming the book didn't specify base 6, or anything as trivially resolvable as such partial transformation for what on the face of it is "obviously wrong". (Of course, "assuming" things is one of our greatest strengths and one of our greatest weaknesses.)
It's rather curious that 41(base10) is also prime do you think?
prime 17(base10)=25(base6)
non-prime 25(base10)=41(base6)
prime 41(base10)=105(base6)
Is that sort of behaviour common (-er than random distribution would suggest,) with alternate translation of number and quantity do you know? it's a shame neither 105 or 253 are prime, but it would be interesting to see the distribution.
But the explanation of how primes are factored (which is of course totally correct in mathematics as we use it,) surely assumes that the grid is of squares.
Might bots or aliens not prefer triangular/pyramidal roots, and determine primes by reference to triangular grids? Intelligent bees might even use hexagonal roots if Man were to die out and bees considerably evolve to become the dominant intelligence on Earth.
We choose squares because our minds find them easier to manipulate than other shapes. As if we had square holes in our minds that best suit square pegs. But if an alien had triangular holes, or bees had hexagonal holes, square tables would look ridiculous, and give all the wrong answers. Of course, that has some pretty heavy-duty implications as to the nature of potential non-human numbers in themselves, and not just in how they're used!
Not strictly relevant to the above, but an indication of why we should perhaps be wary of the "obvious" perhaps:
When the wheel was first invented it took a few thousand years to iron out the bugs. The wheels made by Seller-of-Used-Rocks were square, and all of his customers complained about the bumpy ride. One day his best customer, Basher-of-Small-Furry-Animals, was visiting his workshop. Seller-of-Used-Rocks proudly displayed his newest invention. "Here, Basher, is my new improved wheel. I call it the 'rolleasy'. Isn't she a beauty? Ten skins I'm going to charge; but to you, my friend,a real bargain at six skins each, provided you buy two pairs and a spare."
Basher-of-Small-Furry-Animals stared at the new wheel with some puzzlement. Eventually he said, "But it's triangular!"
"Of course," replied Seller-of-Used-Rocks.
"How can that be an improvement?"
"Don't you see? It's obvious!" enthused Seller-of-Used-Rocks. "One less bump."
Yes, I meant 41 in base 6. 
Regarding the idea of triangular roots, if you look at "perfect triangles," you'll find that they are identical to perfect squares. I'd demonstrate, but I can't post pictures here.
Regarding the idea of triangular roots, if you look at "perfect triangles," you'll find that they are identical to perfect squares. I'd demonstrate, but I can't post pictures here.
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