The sentence below this one is false.
The sentence above this one is true.
What can we now say about the two sentences above? That, if we dispense with the false true/false dichotomy, neither is either true or false: each is true/false or paradoxical. Parodoxes are inevitable in any human language, which in my thinking includes the language of mathematics and all the sciences. It seems to me they never occur in nature. That is, nothing in reality except our attempts to describe is ever paradoxical. Hence, two parallel lines can meet each other in Non-Euclidean geometry, but not in the real world.
An aside: I’ve always wondered at the fact that many intelligent people with scientific backgrounds find the idea that parallel lines that meet will exist if the universe were the surface of a sphere. Only in geometry. In the real world nothing that has less spatial dimensions than two can exist. Start drawing a line on your universe as the surface of a sphere and your first tiny dot with burst out if it on both sides.
Related to this absurdity is the idea of curved space. But that’s just an absurdity of expression–saying, for instance, that a curving ray of light curves due to the effect on space of gravity rather than due to the effect of gravity on it.
It applies, too, to the idea of transfinite numbers. The idea behind that, according to George Gamow is that you can’t form a list of infinitely-repeating decimal fractions that contains them all as shown by the fact (and this is true) that if someone brings up a fraction he claims is not on your list, giving you the first ten digits, say, and you show him it is on your list, he can show it isn’t by showing you the eleventh digit on his is different the eleventh digit on yours, or–in the one in ten chance thatthe digits match, that some other later digit on his fraction is different from the one in that spot on yours.
But that procedure can work oppositely: you can make a list of all possible decimal fractions including infinitely-repeating ones and challenge him to find an infinitely-repeating decimal fraction of a given length that’s not on your list. If you take all the integers in order, reversed, and make them infinitely-repeating decimal fractions by putting a point in front of each one as here:
.1 .2 . . . .9 .01 .11 .21 . . . .91 .02 . . . .99 .001 . . .
he will not be able to give you any decimal that you can’t immediately indicate the location of on your list of a decimal fraction matching whatever string of digits he gives you. Say it’s 00958746537 . . . That will be the 73564785900th number on your list. He can’t then say his number’s twelfth digit is different from your number’s because you can honestly say you don’t know what your number’s twelfth digit is. But that his second number can be as easily found as his first.
So you have two true statements about mathematics that contradict each other. I say they are therefore true/false statements, or paradoxes.
I don’t believe any real object can be infinite. Space may be said to be, but it doesn’t exist, in my philosophy. But I doubt that there’s any way of testing the truth of what is just my opinion.
Note to anyone concerned about my health: I managed to get through a stress test yesterday without having a heart attack or stroke. I’m now wearing a heart monitor that will come off in three hours, at ten a.m. I felt peculiar a few times while wearing it. I hope they don’t turn up as problems on the monitor’s record.
I had trouble walking properly on the stress test treadmill. My doctor and the woman overseeing me got quite frustrated. I felt incredibly stupid. I think I know what happened: there are rails to hold onto that I couldn’t grip without slightly stooping because I’m tall but have the arm-length of a 5′ 9″ white man, so I felt very awkward–until I just let my fingers touch the rails. I also tried too hard because I thought I was going to be tested with stress. When it was over, I asked why they’d never speeded it up. No stress at all except the stress of feeling physically incompetent.
Hmmm, my spell-checker doesn’t like “speeded up.” But surely “sped up” is not correct. Weird. I’d automatically say, “I sped home,” not “I speeded home.” But that’s great. The language should be crazy–as long as it tries for maximal stability.
I have opined on occasion that people who can remember rules of usage like sped/speeded may be use them to show their superiority to those who can’t–in other words, the lay/lie distinction that I hold to, for example, I hold to partly to show I’m not low-class. But I hope my main reason is that I like ways to break out of universal rules as a literary artist. I do believe in social classes, though. It slightly simplifies relating to people, which is incredibly complex. Not that my believing in it matters since a classless society is impossible.
This entry was posted on Thursday, April 21st, 2011 at 12:00 AM and is filed under Philosophy. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
endwar 
Glad you are here, Bob. Good luck on your VENTURE to SC. I hope it’s good. I love science, but not quite so much as I love your work. There you have it. Infinitely verifiable
[…] Entry 121 — Definition of Scientific Account 2 days […]
You may be interested in the following URL: http://www.sciencebuddies.org/science-fair-projects/project_scientific_method.shtml which explains the steps involved in the scientific method. My point being that a scientific account of something is only arrived at through proper use of the scientific method, as out lined in the link. (That link is simpler than the Wikipedia explanation or the Caltech intro to Scientific method:
http://web.ipac.caltech.edu/staff/jarrett/talks/LiU/scien_method/AppendixE.html.
Connie, without really thinking about it (I’m kinda tired right now) I would say that probably following the standard scientific method would result in a scientific account of something, but I claim (violently!) that one can come up with a scientific account of something without an experiment–or, at least without performing an experiment. That is, one can use one’s empirical knowledge to construct a scientific account of something, with one’s empirical knowledge including knowledge of various experiments already performed, informally as well as formally. For instance, I can theorize that men get excited at football games without hooking up ten thousand football fans in a football stadium to blood pressure machines. I think that the obsession with experimentation in science has held it back, and that experimentation rarely leads to anything very significant. Einstein, I think, did few or no experiments.
But thanks for the interest!
–Bob (late with this because only today finding out this blog was getting comments and how to deal with them)
The (violently!) remark suggests you won’t change your mind. Einsitein actully perfomed “thought experiments.” Mathematical proofs and theoretical physics are different, but in the fields of biological and physical science, the only way to confirm a theory is through experimentation. And the development of pennicillin, vaccines and cancer chemotherapy, to name a few, are indeed significant. Regarding your football example, you actually followed the scientific method by hypothesizing that men get excited, testing that hypothesis by reviewing previous data and using observable criteria (instead of BP) to measure excitement (eg., any memory of a game where men stood up and cheered).
Hi, Connie.
> The (violently!) remark suggests you won’t change your mind.
It does, but the suggestion is wrong. That I tend to violently defend certain of my views doesn’t mean I won’t change my mind about them.
> Einstein actually perfomd “thought experiments.”
Sure, but I was referring to physical experiments.
> Mathematical proofs and theoretical physics are different, but in the fields of biological and physical science, the only way to confirm a theory is through experimentation.
That would depend on your definition of experiment. I claim that formal experiments are often unneeded, and by “experiment,” I mean formal physical experiment. Not that such experiments are not often extremely useful, and in some cases necessary. But, just as one can construct a scientific account of something without experiments by basing it on one’s experience of the past, and one’s thoughts about that, one can confirm a hypothesis (which is different from a scientific account) by checking it against the past and thinking about it. Also by further investigation, which is similar to experimentation but not really experimentation, for me. For instance, I hypothesize certain brain mechanisms. One needn’t set up an experiment to find out if they exist, one need only keep investigating the brain physically, or even research old investigations. I admit that this is hair-splitting. Obviously, if your scientific account is of something not available to normal sensory experience, you need to experiment or do something close to it to validate it.
In many fields, particularly what you might call macroscopic psychology, the experiments have been done in the real world. We don’t have to do a formal experiment to find out if children will go to a man handing out free ice cream cones on a hot day or to a man handing out religious pamphlets.
> And the development of penicillin, vaccines and cancer chemotherapy, to name a few, are indeed significant. Regarding your football example, you actually followed the scientific method by hypothesizing that men get excited, testing that hypothesis by reviewing previous data and using observable criteria (instead of BP) to measure excitement (eg., any memory of a game where men stood up and cheered).
Right. No formal experimentation necessary. Empirical knowledge necessary.
thanks for your response, Bob