Philosophy Mathematical Notion Of Infinity Essays -

Philosophy: Mathematical Notion Of Infinity The mathematical notion of infinity can be conceptualized in many different ways. First, as counting by hundreds for the rest of our lives, an endless quantity. It can also be thought of as digging a whole in hell for eternity, negative infinity. The concept I will explore, however, is infinitely smaller quantities, through radioactive decay Infinity is by definition an indefinitely large quantity. It is hard to grasp the magnitude of such an idea. When we examine infinity further by setting up one-to-one correspondence's between sets we see a few peculiarities. There are as many natural numbers as even numbers. We also see there are as many natural numbers as multiples of two. This poses the problem of designating the cardinality of the natural numbers. The standard symbol for the cardinality of the natural numbers is o. The set of even natural numbers has the same number of members as the set of natural numbers. The both have the same cardinality o. By transfinite arithmetic we can see this exemplified. 1 2 3 4 5 6 7 8 ? 0 2 4 6 8 10 12 14 16 ? When we add one number to the set of evens, in this case 0 it appears that the bottom set is larger, but when we shift the bottom set over our initial statement is true again. 1 2 3 4 5 6 7 8 9 ? 0 2 4 6 8 10 12 14 16 ? We again have achieved a one-to-one correspondence with the top row, this proves that the cardinality of both is the same being o. This correspondence leads to the conclusion that o+1=o. When we add two infinite sets together, we also get the sum of infinity; o+o=o. This being said we can try to find larger sets of infinity. Cantor was able to show that some infinite sets do have cardinality greater than o, given 1. We must compare the irrational numbers to the real numbers to achieve this result. 1 0.142678435 2 0.293758778 3 0.383902892 4 0.563856365 : : No mater which matching system we devise we will always be able to come up with another irrational number that has not been listed. We need only to choose a digit different than the first digit of our first number. Our second digit needs only to be different than the second digit of the second number, this can continue infinitely. Our new number will always differ than one already on the list by one digit. This being true we cannot put the natural and irrational numbers in a one-to-one correspondence like we could with the naturals and evens. We now have a set, the irrationals, with a greater cardinality, hence its designation as 1. Georg Cantor did not come up with the concept of infinity, but he was the first to give it more than a cursory glance. Many mathematicians viewed infinity as unbounded growth rather than an attained quantity like Cantor. The traditional view of infinity was something ?increasing above all bounds, but always remaining finite.? Galileo (1564-1642) noticed the peculiarity that any part of a set could contain as many elements as the whole set. Berhard Bolzano (1781-1848) made great advancements in the theory of sets. Bolzano expanded on Galileo's findings and provided more examples of this theme. One of the most respected mathematicians of all time is Karl Friedrich Gauss. Gauss gave this insight on infinity: As to your proof, I must protest most vehemently against your use of the infinite as something consummated, as this is never permitted in mathematics. The infinite is but a figure of speech; an abridged form for the statement that limits exists which certain ratios may approach as closely as we desire, while other magnitudes may be permitted to grow beyond all bounds....No contradictions will arise as long as Finite Man does not mistake the infinite for something fixed, as long as he is not led by an acquired habit of mind to regard the infinite as something bounded.(Burton 590) Cantor, perhaps the true champion of infinity, built off of his predecessors findings. He argued that infinity was in fact ?fixed mathematically by numbers in the definite form of a completed whole.?(Burton 590) Cantor looked to

Planting, Growing, and Marketing Royal Paulownia

Planting, Growing, and Marketing Royal Paulownia Paulownia tomentosa has had marvelous press on the Internet. Several Australian and United States companies make claims of extraordinary growth, unbelievable wood values, and magnificent beauty. Paulownia, they write, can shade an area in record time, resist insects, feed livestock, and improve the soil component - and in some ways this is correct. But is this just hype or is the plant truly a supertree Let me introduce you to Royal Paulownia and you just might rethink the abilities given to the tree by producers. Empress Tree - Mythology vs. Facts You can tell this tree is very special right away, from just its name. The plants pedigree and regal names include Empress Tree, Kiri Tree, Sapphire Princess, Royal Paulownia, Princess Tree, and Kawakami. The surrounding mythology abounds and many cultures can claim title to embellishing the plants many legends. Many cultures love and embrace the tree which in turn promoted its worldwide popularity.  The Chinese were the first to establish a much-practiced tradition that included the tree. An oriental Paulownia is planted when a daughter is born. When she marries, the tree is harvested to create a musical instrument, clogs or fine furniture; they then live happily ever after. Even today, it is a valued wood in the orient and top dollar is paid for its procurement and used for many products. A Russian legend has it that the tree was named Royal Paulownia in honor of Princess Anna Pavlovnia, daughter of Russias Czar Paul I. Its name Princess or Empress tree was an endearment to a nations rulers. In the United States, many of these trees have been planted for wood production  but naturalized wild stands grow along the Eastern Seaboard and through the mid-western states. Paulownias range is said to have expanded because of the seed pods used in packing shipped cargo from China early in the last century. Containers were emptied, winds scattered, the tiny seeds and a fast paulownia forest developed. The tree has been in America since introduction during the mid-1800s. It was first discovered as a profitable tree in the 1970s by a Japanese timber buyer and the wood was purchased at attractive prices. This sparked a multimillion-dollar export market for the wood. One log is said to have sold for $20,000 US dollars. That enthusiasm has mostly run its course. One thing to remember is, the wood is totally ignored by domestic timber companies in the United States and speaks volumes about its economic potential, at least to me. But utilization studies by several universities including Tennessee, Kentucky, Maryland, and Virginia suggests the potential  for a favorable future market. Should You Plant Royal Paulownia? There are some compelling reasons to plant Paulownia. The tree has some of the best soil, water, and nutrient retaining properties. It can be made into forest products. At first blush, it makes sense to plant Paulownia, watch it grow, improve the environment, and make a fortune at the end of ten to twelve years. But is it really that simple? Here are the attractive reasons for growing the tree: Paulownia is a light, air curable wood, that does not warp, twist, or crack. The tree is fire resistant and water repellent. These are very good wood qualities and the tree has all of these. Paulownia can be sold for pulp, paper, poles, construction material, plywood, and furniture and at top dollar. You still have to be lucky enough to be growing the trees in an area with a good market. Paulownia can be commercially harvested in five to seven years. This is true but only for some products made by companies that may or may not be buying at any given time. Paulownia is a beautiful tree and is easily propagated from root cuttings. But it can also become a problem in the landscape because of its messy habits. Paulownia is nitrogen rich and makes an excellent livestock fodder and soil amending mulching material.   If all of these statements are true, and for the most part they are, you would be doing yourself a favor to plant the tree. It would, in fact, be a great idea to plant the tree on a good site. Great for the environment, great for shade, great for soil, great for water quality and great for a beautiful landscape. But is it economically sound to plant Paulownia over large areas? Are Paulownia Plantations Economically Practical? A recent discussion on a favorite forestry forum was are Paulownia plantations economic? Gordon J. Esplin writes promoters of Paulownia plantations are claiming incredible growth (4 years to 60, 16 at breast height) and value (eg $800/cubic meter) for Paulownia trees. This seems to be too good to be true. Are there any independent, scientific studies on the species? James Lawrence of Toad Gully Growers, a Paulownia propagation company in Australia sums it up completely. There has, unfortunately, been much over-hyped promotion of Paulownia. It is true, however, that under the right conditions, Paulownia produces valuable timber in a shorter time frame... Lawrence goes on to say that it usually takes from 10 to 12 years to achieve a size economical to mill and is not construction strong enough to be used as building material. It is most likely to find its place in moldings, doors, window frames, veneers, and furniture. He further says that trees in the cooler regions of Australia may be more slowly grown and consequently of higher timber quality - close growth rings are desired for furniture - than those grown in warmer climates; however, the higher rate of crop rotation in the warmer zones should compensate for any lower returns per m3. Lawrence just indicated, at least to me, that we need to take a deep breath and grow the tree slower for optimum quality. And what about a little thing called market? Remembering that the top three things that affect the value of any real property are location, location, location, I would suggest that the top three things that affect the value of standing timber price are markets, markets, markets. Paulownia is no different from any other tree in this regard and you need to find a market before planting and I have found no support for a market on the Internet. The literature suggests that the present US market is extremely under-developed in Paulownia and one source actually suggested that there is no present market. The future of this tree depends on a future market. I did run across a credible reference to price. Mississippi State University indicates in a report on Unique Species and Uses that Paulownia logs have been found growing in the Mississippi Delta and south along the Mississippi River. Paulonia logs have been in high demand in Japan and bring excellent prices (my emphasis) to landowners in Mississippi. I have yet to find that buying source. Also, there are risks associated with any tree planting venture. Paulownia is no different. It is sensitive to drought, root rot, and diseases. There is also the economic risk of producing a tree with little future economic value.