Seasons

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...

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"?

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.

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.

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.

"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.

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?

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.