Formal Logic and Scientific Method

nderosa,

This is a very good analysis and I agree with almost all of it. I think though that ideas which require a suspension or violation of logic should always be excluded from consideration no matter whether the opposing ideas can ever be factually verified or not. It makes no sense to give illogical ideas the same standing as logical ones. Illogical ideas can never be shown to be factually true, so there is no point in considering them.


JR

Good. Ok, let's go back to the "minimun length" issue with the above in mind.
In the Zeno thread, you restated the argument given by TVF (which was later
rerestated by Astro) for why there can be no minimum length between particles
in space (besides 0)

In logical format the argument is as follows:
1. We start with the assumption of a minimum length between particles in space.
2. TVF argues that in a system of 3 particles, if one is placed at the minimum
length from the central particle and a second one is also placed the minimum length from the central particle but at an angle from the first such that the distance between the outer particles would be less than the minimum length, the distance between the outer particles would be less than the minimum length.
3. Since 2 contradicts 1, the starting assumption is supposedly reduced to the absurd.

The fallacy of TVF's argument is easily seen if we had attempted to draw the distance between the outer particles first. The
circularity of it is also readily evident by reading 2 aloud.

If our starting assumption is that there is a minimum length between particles (besides
0), it is implicit in the assumption that not all angles between particles are allowed-
specifically, those angles that would violate our starting assumption of a minimum
length between particles in space. DR. Flandern's refutal is like if we had discovered that all pentagans only have 5 sides, he draws me a square, then uses that to disprove the factual statement. Obviously irrelevant.

Just as a universe in which two parallel lines don't meet requires a different geometry than one that does, the geometry in one that has a minimum length will obviously differ from
one that doesn't. Here's a simple illustration. Draw a 3 by 3 square, divide it into
9 1X1 squares. If each square represents the minimum area of particles and divisions
of space, then referring to the 3 particles in TVF's argument, it's impossible to obtain
a configuration in which the three particles are less than the minimum length of one
side of the minimum squares apart, without being adjacent to each other.

And interestingly enough, in QM's model of the hydrogen atom (which is based on countless experiments) angular momentum is quantized. The electrons in the atom do not orient
in every possible angle in space. Spectroscopy also shows that energy is in some way
quantized. With all these and countless other empirical facts suggesting that our starting
assumption of a minimum length is a good one, I think it should take a little more
than one simplistic argument, especially one that is wrong, for any reasonable person
to dismiss it.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">If our starting assumption is that there is a minimum length between particles (besides 0), it is implicit in the assumption that not all angles between particles are allowed-specifically, those angles that would contradict our starting assumption of a minimum length between particles in space. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Actually, I thought of this same idea when I first read "Dark Matter,..." only a few months ago, but I rejected it.

It would require that all angles be multiples of 60 degrees (pi/3 radians). This is contradicted by the fact that we have angles that are not multiples of 60 degrees (the 105 degree angular separation of the hydrogen atoms in a water molecule, for example).

There is also another problem with your suggestion. Suppose we have four particles <i>A</i>, <i>B</i>, <i>C</i>, and <i>D</i>, arranged as below:

-o
o o
-o

Particle <i>A</i> is the top one, <i>B</i> is the leftmost one in the second row, <i>C</i> is the rightmost one in the second row, and <i>D</i> is the bottom one. The hyphens are there only to make the diagram look like I wanted it to--please ignore them. As in my other example in the other thread, I'll refer to the minimum possible distance as <i>d</i>.

Angle <i>BAC</i> would be a 60 degree angle, as would angle <i>ABC</i>, <i>ACB</i>, <i>CBD</i>, <i>BCD</i>, and <i>BDC</i>. This allows the distance between all adjacent particles to be <i>d</i>--the distance between <i>A</i> and <i>B</i> is <i>d</i>, as is the distance between <i>A</i> and <i>C</i>, <i>B</i> and <i>C</i>, <i>B</i> and <i>D</i>, and <i>C</i> and <i>D</i>. Furthermore, angles <i>ABD</i> and <i>ACD</i> are both 120 degree angles, which is in agreement with the supposition that all angles be multiples of 60 degrees (120=2*60).

But what is the distance between <i>A</i> and <i>D</i>? If we do the geometry, we find that the distance is <i>d</i>*SQR(3), or approximately 1.732<i>d</i>.

So we see the same problem as before: We end up with a distance that is not a multiple of <i>d</i>, which suggests that <i>d</i> itself is divided into smaller units (in fact, we get an irrational number, which extends for an infinite number of digits, implying that <i>d</i> is infinitely divisible), contradicting our original assumption that <i>d</i> is the smallest possible distance.

No matter what, if you assume a minimum possible length, there will always be some possible arrangement of particles that contradicts that assumption.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">If each square represents the minimum area of particles and divisions of space, then referring to the 3 particles in TVF's argument, it's impossible to obtain a configuration in which the three particles are less than the minimum length of one side of the minimum squares apart, without being adjacent to each other. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

What about the distance between the opposite corner squares of your 3x3 square? The distance between the top left square and the bottom right square would be approximately 1.414 times the length of the side of one of your minimum squares; the same is true of the distance between the top right square and the bottom left square. Which would mean that your minimum squares are not really minimum squares at all.

I can't picture any possible way in which the assumption of a minimum possible length will not lead to self-contradiction of that assumption.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by 1234567890</i>
<br /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Astrodelugeologist</i>
<br /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">If our starting assumption is that there is a minimum length between particles (besides 0), it is implicit in the assumption that not all angles between particles are allowed-specifically, those angles that would contradict our starting assumption of a minimum length between particles in space. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Actually, I thought of this same idea when I first read "Dark Matter,..." only a few months ago, but I rejected it.

It would require that all angles be multiples of 60 degrees (pi/3 radians). This is contradicted by the fact that we have angles that are not multiples of 60 degrees (the 105 degree angular separation of the hydrogen atoms in a water molecule, for example).

<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

The further apart the
particles are, the more angles of connection become available to
them. It really depends on the relationship between 2 of the 3 particles- any angle that doesn't cause the distance between two
of all 3 pairs to be less than the minimum would be allowed.

There is also another problem with your suggestion. Suppose we have four particles <i>A</i>, <i>B</i>, <i>C</i>, and <i>D</i>, arranged as below:

-o
o o
-o

Particle <i>A</i> is the top one, <i>B</i> is the leftmost one in the second row, <i>C</i> is the rightmost one in the second row, and <i>D</i> is the bottom one. The hyphens are there only to make the diagram look like I wanted it to--please ignore them. As in my other example in the other thread, I'll refer to the minimum possible distance as <i>d</i>.

Angle <i>BAC</i> would be a 60 degree angle, as would angle <i>ABC</i>, <i>ACB</i>, <i>CBD</i>, <i>BCD</i>, and <i>BDC</i>. This allows the distance between all adjacent particles to be <i>d</i>--the distance between <i>A</i> and <i>B</i> is <i>d</i>, as is the distance between <i>A</i> and <i>C</i>, <i>B</i> and <i>C</i>, <i>B</i> and <i>D</i>, and <i>C</i> and <i>D</i>. Furthermore, angles <i>ABD</i> and <i>ACD</i> are both 120 degree angles, which is in agreement with the supposition that all angles be multiples of 60 degrees (120=2*60).

But what is the distance between <i>A</i> and <i>D</i>? If we do the geometry, we find that the distance is <i>d</i>*SQR(3), or approximately 1.732<i>d</i>.

So we see the same problem as before: We end up with a distance that is not a multiple of <i>d</i>, which suggests that <i>d</i> itself is divided into smaller units (in fact, we get an irrational number, which extends for an infinite number of digits, implying that <i>d</i> is infinitely divisible), contradicting our original assumption that <i>d</i> is the smallest possible distance.

<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

I have not fully considered the additional assumption of requiring
distances to be integer multiples of the minimum length , but if that was the case then your arrangement of particles would be impossible since it would violate that assumption. I also don't believe that the presence of lengths not in integer multiples of
the minimum distance logically contradicts the arguments I made.
If all papayas are found to be 6 inches or more in diameter and I found one that is 6.5 inches in diameter, does it follow that
0.5 inch papayas must exist? No, I don't think so.

But I do agree that quantities existing in integer multiples of the minimum allowed is a more appealing idea. Is such a geometry possible? I don't see why not. And just like the idea of parralell lines crossing is not self contradictory when viewed
under a non-Euclidean geometry, the idea of a wholly quantized
geometry is not self-contradictory in the context of its own axioms.
Note that we don't question the self-consistency of natural numbers, even though the whole set is based on the idea of having a minimum length and multiples of it. Namely, 1.

The way I look at it is if you have something sticking out of the particles that are at least a certain minimum length, then that's the minimum length possible the particles can get. And if these things that stick to the particles are only available in the same sizes, then you can only add more of them together in multiples of this length. And the resulting geometry of particles in space would be determined by these physical parameters.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by nderosa</i>
<br />Many of the participants of this message board seem to understand that a conclusion properly drawn in a syllogism is necessarily true.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The way I was taught logic, if the reasoning is valid, then the conclusion drawn in a syllogism is as reliable as the premises it is based on. That is very different from claiming it is "necessarily true". Much of your message belabors this same point.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">This leads me to conclude that a theoretical over reliance on deduction is a weak link in any scientific method, and not essential to the actual process of acquiring scientific knowledge, which relies on both deduction and induction as it properly should.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I argued in <i>Dark Matter …</i>, p. 362 ff, that both induction and deduction have a role, but that only deduction should be used (when it is available) for developing theories because induction is non-unique and rarely better than educated guesswork. With deduction, because the process is unique, it would be an unlikely happening for the theory to predict the observations well unless the starting point was in fact valid.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">To me, the principle of economy (Ockham's razor) dictates that we keep things as simple as possible.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">"Simple" meant "fewest assumptions" or "fewest degrees of freedom" to Ockham. One must be very wary of associating one's prior beliefs with "simpler" because that is a form of bias. Old comfortable knowledge always seems simpler at first compared with strange new ideas because often, many consequences depend on this choice, and it takes a long time to sort through them all. The world was once comfortable with Ptolemy's epicycles and wary of Newton's universal law of gravitation because the latter was too strange and unfamiliar and obviously could not explain an Earth-centered universe, which everybody at the time "knew" to be true. Many Middle Ages people would have called Ptolemy's theory "simpler" than Newton's. But it is not simpler in the Ockham sense.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">-A "smallest possible particle" hypothesis is a common sense notion based on everyday experience. It may be factual or an imaginary construct. There is insufficient data to know for sure.
-An infinitely (or indefinitely) small particle or distance is a mathematical construct which may be factual and may be a figment of the imagination. There is insufficient data to know for sure.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">"Insufficient data" is a red herring because no amount of data could ever demonstrate infinite divisibility. As Neil says, "based on everyday experience", everything has its own smallest units. And just as for Ptolemy's epicycles, that seems more familiar and comfortable. But in no valid sense is it simpler. To determine that, we need to deduce the logical consequences of both hypotheses. Infinite divisibility and one-to-one correspondences lead to descriptions similar to reality without further complications. However, "smallest possible particle" leads to a dozen paradoxes such as those I listed from <i>Dark Matter…</i>, each of which seems to need a new hypothesis to explain. So "smallest possible" fails by the "simpler" criterion as described by Ockham, even if not by the "familiar and comfortable" criterion. -|Tom|-

The MM was presented to me as a deductive model. It is seductive to the mind not due to the soundness of the logic (logic is man's creation). The model seems right because it leads to what we see. Observation in the scientific method is often harped about but less often done. I state this because ideas like uncertainty are still hotly debated, yet think of all the observations made since then. They don't fit the accepted theories, yet change is so hard that the swidgets are in place AS fact! The logic isn't faulty in science because the scientific method is NOT being used in science, at universities, or in the media. We cannot have half science. It is all or nothing. Mark