Entry 344 — Reason Can Explain Everything
Too worn-out today, who knows why, to have anything for this entry but an opinion. It’s the one in my title. I believe that every question will result in one of three answers: a rational answer, a failed attempt at a rational answer, and a willfully false answer. The first will come about when someone intelligent uses reason on a question that proves tractable. The second will come about either when person of limited intelligence is unable to use reason on a tractable question, or when an intelligent person uses reason on a question that proves untractable–because the person lacks the intelligence or knowledge to deal with it. The third will come about when a person consciously or unconsciously needs it to be to his liking much more than he needs it to be true, so he does not use reason on it.
A good example of the latter is the answer of many to the question, “What is poetry?” Some poets need this answer to be “something too sublime to be defined,” because that seems to them to raise them to the level of supreme priests of some sort, wise in the ways of secrets beyond the ken of the uninitiated. Others need it bo be “something too vaguely defined not to apply to just about anything,” which allows anyone to call himself a poet, which will gratify a egalitarian, and win him followers since those happy with criterialessness are always much more numereous than those who are not. Finally, many will need it to be beyond reason because they are deficient reasoners, so don’t want reason to be consider of any real value.
I’ve left something out of this discussion: the fact that ost people concerned with the question of what poetry is, are really concerned with the question of what a moving poem is. It is that which they claim reason can’t begin to explain. But I am sure it can be. I feel fairly confident that I’ve done it–essentially as something that causes pleasure in certain inter-related parts of the brain, a sort of poetry center that neurophysiologists will eventually pin down. I have detailed ideas as to exactly what will cause that center to experience pleasure, too: in brief, a text, with or without averbal matter, that qualifies as a poem by my definition (i.e., a text with a certain percentage of flow-breaks), that is neither too familiar or too unfamiliar to the person encountering it.
Similarly other supposedly beyond-reason things like love and hatred can be similarly explained rationally by the existence of love and hatred centers.
That it is sometimes extraordinarily difficult to find an answer to a question does not mean that reason will never find an answer to it.
Bob,
I don’t think your transfinite number statements make sense. Basically, you can show that the infinite number of fractions is equal to the infinite number of integers is equal to the infinite number of positive whole numbers, by showing a way to map them all. But you can also show that the (infinite) number of real numbers is greater than the infinite number of integers.
the way to show that that the number of fractions (or rational numbers) is to find a way to list them so that you can count them all, so that then given a fraction you can figure out what number it is on your list of fractions. Since all fractions are written as ratios of numbers, you can write them like this:
1/1 1/2 1/3 1/4 1/5 . . . .
2/1 2/2 2/3 2/4 2/5 . . . . .
3/1 3/2 3/3 3/4 3/5 . . . . .
4/1 4/2 4/3 4/4 4/5 . . .
5/1 5/2 5/3 5/4 5/5 . . .
Then you can count by starting with at the corner with 1/1 and then moving around to 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, and so on. Now you are counting them in a pyramid, and you can label the rows of the pyramid by the sum of the numerator and denominator, so the first pyramid row consists of 1/1, which we label row 2 because 1+1=2. Then there’s row 3 with two members 1/2 and 2/1, and for each 1+2=2+1=3. The next row is 4 with 3 members, and 1+3=2+2=3+1=4. Now if you give me any (positive) fraction a/b, i can compute it’s position in my pyramid scheme — it is the a-th term in row (a+b), so it’s number is the Sum of 1+2+3+4+. . . +(a+b-1)+a. In other words I count up the terms in the preceding rows in my pyramid and add a because that’s where it is in the last row of fractions. So there’s a way to count fractions, which means that the total number (that degree of infinity) is the same.
But the number of real numbers is greater, and the proof works this way. Suppose you have some scheme for counting real numbers (we can even limit it to real numbers between 0 and 1, if you like), then take your first number on your list and write it as a decimal. If the decimal repeats, fine. If it ends, you can always add 0s on the end, after all 0.25=0.250=0.25000=0.25000000000000000000, etc.
Then i can take your list and generate a new decimal number that isn’t on your list. Suppose the first number starts 0.1, so i pick a different digit than 1, say 2,and start my new number as 0.2. If your next number starts 0.037763902, then i look at the second digit, 3, and pick a different digit, say 2, and so my number 0.22 . . . isn’t the second one on your list either. For the third digit, i look at the third digit of your third number and pick something different (I have 9 choices), for the fourth digit i look at the fourth digit of your fourth number and pick something different there. If you would prefer an algorithm to pick each digit, you can always add 1 cyclically, so that if the nth digit of your nth number is 9, i pick 0 for my nth digit. So thus you see that for any counting scheme you have for real numbers, i can make a number that isn’t on your list.
You can also show that the infinite number of points (real numbers) between 0 and 1 is equal to the number of points greater than one, because you can draw a one-to-one correspondence between x and 1/x.
Infinite numbers are pretty crazy, but there are actually rigorous ways of working with them.
– endwar
Cantor wants a list of decimal fractions you can count. I don’t. I just want a list that every decimal fraction will eventually show up on, and I have it. The fact that when I find some decimal fraction asked for on my list, a second decimal fraction can be named of greater length that may not be the number I’ve just found on my list is lrrelevant. Bringing in non-repeating decimals is just a con game like bringing in lies and non-lies into linguistics to produce paradoxes. I’m sure transfinite numbers are fun for mathematicians but also sure they are inapplicable to anything in the real world. According to my philosophy. (And my neurophysiology, which holds that numbers are real secondary characteristics of real things, and that–possibly–addition and subtration and may be further processes are real the way motion is. I’m working on an attempt coherently to show how this is that may take me a while because it’s currently low on my list of priorities.)