The Concept of Infinity
Consider Zeno's Dichotomy Paradox. To go anywhere, say out the nearest door, you must first go half the distance. Then half the distance left. There are an infinite number of half distances between you and the door, therefore you can never get through. The mathematical formula for it is simple:
∞ n
∑ (1/2) = 1/2 + 1/4 + 1/8 + 1/16 + . . .
n=1
There are many attempts to explain how you can get out the door, which you obviously can, or you wouldn't be here to read this.
The silliest was on the site where I found the formula. The writer just blithely comments "it equals 1." Period, case closed. But it demonstrably does not equal 1, for it is always that 1/2 distance left short of 1, hence the part to the right of the equal sign does not just read "1".
Then there are those who say that you move faster over the shorter distances. Huh? There is no time factor involved. If you want to take 5 hours going halfway, then 20 years going the next 1/2, it is still half the remaining distance, still infinite, still unreachable.
Lastly, I'll mention the solution put forth by the late, great Tom Van Flandern. While I admire his Meta Model for Cosmology, his attempt is equally unacceptable. He posits the distances we travel being like 1, 2, 3, but the infinity fills the gaps between the numbers (like 1.1, 1.2, 1.3. . .); basically the difference between the Counting Numbers and the Real numbers. We jump from Counting Number to Counting Number. It leaves unsaid, though, how we traverse that Real Number gap, which is itself infinite. So it is no solution, either.
Here's my much more elegant solution: infinity is not a reality, it is a concept. First step in the Scientific Method is Observation. Where in all of Creation do we see anything that is infinite? There is an edge to the Universe, scientists say; there are numberless stars in mind-numbing quantity of galaxies, but not an infinite number; molecules, atoms, quantum particles, all are finite in number, 'though we have no number we can give them.
Infinity is a concept, nothing more.
This is, of course, not intellectually satisfactory, either. We can more than conceive of halfways, we can create them (up to a point, the limitation being our physical technological abilities). Yet we live in an Universe of Time and Distance, not Infinities. So it stands, unresolved.
And an important other point, which will come up in many of my opinion pieces: a mathematical formula may reflect Reality, but Reality is not a reflection of a mathematical formula. Just because the math works, doesn't mean it is real.
Consider this. If you put $500 in the bank, then write checks totaling $1000, according to the math (500 - 1000 = -500) you have -$500. Try getting the bank to give you the -$500. The math works, but the negative amount is not real, it is only, again, a concept.
While on math and Infinity, here is another illogical, irrational, modern "fact" (not!), again based on a mathematical "proof":
Modern Mathematics claims that there are different size Infinities. The Real Numbers are said to be a larger Infinity than the Counting Numbers. Well sure, since they have an infinite number of digits between the Counting Numbers, it must be larger, right?
No, I can still make a 1 to 1 correspondence between the Counting and Real numbers, even though I may never get to 2 in the Real Numbers. For every Real Number you can name, I can name a Counting Number. Ad Infinitum.
But let's assume they are correct, that Infinities can be different sizes. Then how do you explain their other claim that the Counting Numbers are the same size Infinity as the Odd Numbers and the Even Numbers? By definition, the Odd or Even Numbers are 1/2 of the Counting Numbers. If Infinities vary in size, they would have to be 1/2 the size of the Counting Numbers.
Check your logic at the door, please refuse to think rationally, let us tell you what to believe, for we are the wise ones.
Sorry, I won't. . .and I hope you don't.
Consider Zeno's Dichotomy Paradox. To go anywhere, say out the nearest door, you must first go half the distance. Then half the distance left. There are an infinite number of half distances between you and the door, therefore you can never get through. The mathematical formula for it is simple:
∞ n
∑ (1/2) = 1/2 + 1/4 + 1/8 + 1/16 + . . .
n=1
There are many attempts to explain how you can get out the door, which you obviously can, or you wouldn't be here to read this.
The silliest was on the site where I found the formula. The writer just blithely comments "it equals 1." Period, case closed. But it demonstrably does not equal 1, for it is always that 1/2 distance left short of 1, hence the part to the right of the equal sign does not just read "1".
Then there are those who say that you move faster over the shorter distances. Huh? There is no time factor involved. If you want to take 5 hours going halfway, then 20 years going the next 1/2, it is still half the remaining distance, still infinite, still unreachable.
Lastly, I'll mention the solution put forth by the late, great Tom Van Flandern. While I admire his Meta Model for Cosmology, his attempt is equally unacceptable. He posits the distances we travel being like 1, 2, 3, but the infinity fills the gaps between the numbers (like 1.1, 1.2, 1.3. . .); basically the difference between the Counting Numbers and the Real numbers. We jump from Counting Number to Counting Number. It leaves unsaid, though, how we traverse that Real Number gap, which is itself infinite. So it is no solution, either.
Here's my much more elegant solution: infinity is not a reality, it is a concept. First step in the Scientific Method is Observation. Where in all of Creation do we see anything that is infinite? There is an edge to the Universe, scientists say; there are numberless stars in mind-numbing quantity of galaxies, but not an infinite number; molecules, atoms, quantum particles, all are finite in number, 'though we have no number we can give them.
Infinity is a concept, nothing more.
This is, of course, not intellectually satisfactory, either. We can more than conceive of halfways, we can create them (up to a point, the limitation being our physical technological abilities). Yet we live in an Universe of Time and Distance, not Infinities. So it stands, unresolved.
And an important other point, which will come up in many of my opinion pieces: a mathematical formula may reflect Reality, but Reality is not a reflection of a mathematical formula. Just because the math works, doesn't mean it is real.
Consider this. If you put $500 in the bank, then write checks totaling $1000, according to the math (500 - 1000 = -500) you have -$500. Try getting the bank to give you the -$500. The math works, but the negative amount is not real, it is only, again, a concept.
While on math and Infinity, here is another illogical, irrational, modern "fact" (not!), again based on a mathematical "proof":
Modern Mathematics claims that there are different size Infinities. The Real Numbers are said to be a larger Infinity than the Counting Numbers. Well sure, since they have an infinite number of digits between the Counting Numbers, it must be larger, right?
No, I can still make a 1 to 1 correspondence between the Counting and Real numbers, even though I may never get to 2 in the Real Numbers. For every Real Number you can name, I can name a Counting Number. Ad Infinitum.
But let's assume they are correct, that Infinities can be different sizes. Then how do you explain their other claim that the Counting Numbers are the same size Infinity as the Odd Numbers and the Even Numbers? By definition, the Odd or Even Numbers are 1/2 of the Counting Numbers. If Infinities vary in size, they would have to be 1/2 the size of the Counting Numbers.
Check your logic at the door, please refuse to think rationally, let us tell you what to believe, for we are the wise ones.
Sorry, I won't. . .and I hope you don't.