MacroEvolution
Convergent Evolution 
Premordial Soup
Gene Duplication
Transition Fossil 
Posted on 05/18/2005 11:23:17 AM PDT by DannyTN
Either that or A LOOKUP TABLE IN YER HEAD.
(ANYone want to buy a used CAPSLOCK key?)
Then they need to read "Tiling the Plane" for a bit of diversity in their busy, humdrum lives.
Just HOW do you KNOW this information?
As a diversion, go to the park with a friend and some tennis balls.
Get on the rotary spinning wheel thingy (merry-go-round?), sitting directly opposite each other.
Spin the wheel at a good pace, and gently toss the balls between one another.
I will try this one more time. First of all, I do NOT believe bees can do calculus. That is kind of my whole point. Second, your comparison would work if you presented it like this:
You tell me that you are going to throw a ball to me. You tell me the starting height, the angle, the force, wind speed, etc. Then you ask me to place my hand where I think the ball will be so I can catch it but you add the requirement that once I position my hand, I can't move it. If I knew calculus, I could calculate the position fairly accurately and place my hand in the right spot to catch the ball.
That is, essentially, what we are asking the bees to do. We know, a priori, what the displacement has to be in order to minimize surface area (i.e. bee's wax) when constructing the pointy end of the honeycomb. (That's like the placement of the hand in the throwing example.) Then the bees start building and, low and behold, the displacement they build in to the pointy end matches the value we came up with that minimizes the wax used. And, no, bees can't do calculus. That's the WHOLE POINT.
Really, I don't know it. That is why sometimes I drop or miss the ball. Also, the first place I put my hand when the ball is thrown is likely not the final placement when the ball is caught. The position is adjusted, a posterior, based on what I am seeing. To claim the bees are making building decisions in terms of the displacement used, a posterior, would be to imply they are actually doing calculus. Clearly, that is not the case. That is the difference.
So what exactly is your point? When the behavior of bees and termites and birds building nests has been carefully investigated the rules for their construction behavior turn out to be very simple and involve no knowledge or calculations.
You are not going to like my answer so let me ask you a question instead.
Did we invent or discover mathematics? That is the fundamental question.
That is one of those pseudo-questions like, do you believe in free-will or determinism. Good for keeping college kids awake at night, but leading nowhere.
For the record, I lean toward invention for axioms and discovery for the implications of the axioms.
I disagree. It is a fundamental origins question.
For the record, I lean toward invention for axioms and discovery for the implications of the axioms.
Your belief has some problems. Historically, secular mathematicians have agreed that we should have absolutely no expectation that our invented axioms would have any application in the real world. The fact that over and over again they do is what Einstein referred to as a "miracle".
Just like my vintage 2002 CD player - it's still flashing "1200". I guess it's 12 noon somewhere.
Axioms are selected because they are effective in the real world. But when the boundaries of the real world expand due to experience, the axioms begin to fray at the edges.
We cannot imagine violations of the fundamental axioms of arithmetic because we have no experience with exceptions. Arithmetic is an extension of counting, and we do not experience anomolies in the process of counting.
We have experienced anomolies in geometry, leading to adjustments in the axioms of geometry. Imagine trying to explain to the ancient Greeks that the geometry of relativity describes the real world.
So that's where my watch evolved from, and all the time I thought it was made by Timex.
You make mathematics sound like a dart-board approach. Throw out everything and keep what sticks. In my experience, that is not how mathematics is developed. But, anyway, such a posterior axioms are not what I am talking about.
Where the heart of the matter lies is in two areas. First, in purely theoretical mathematics and physics and, second, in very small or invisible natural processes that we have been able to describe mathematically before we could observe them.
In the first case, effectiveness in the real world is not a consideration because the axioms are developed in the purely theoretical realm. In the second case, effectiveness in the real world is not a consideration because it was only later, after the development of the axioms, that their effectiveness in describing the real world was realized once we had developed the technology to test them.
Now, lest you propose that a whole bunch of axioms are developed and tested and we keep the ones that work, I will relate to you this story to show that, actually, the opposite is true.
There was a mathetician in the 20th century who found the whole idea that mathematics is discovered so offensive that he purposely set out to develop theoretical mathematics that would have no application in the real world. After his death, his work turned out to be useful in genetics and in the study of temperatures in smelting furnaces.
MacroEvolution
Convergent Evolution
Premordial Soup
Gene Duplication
Transition Fossil
and then there spider webs....
Yeah....
Everywhere along W30° 9' 47"
I
Daylight savings time!
Question for the hour:
How can we tell that mechanical dial clocks were invented in the Northern hemisphere?
If a bacterium can have a clock, a bee could easily have a miniature computer. I've heard they like Apples.
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