Formal Logic and Scientific Method

<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">What about any right triangle whose integer squared sums yield a hypotenuse with another integer squared sum? Like a 3, 4, 5
right triangle for example?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Ok, that one would work. (And some time after I made that post, I realized that four particles arranged in a tetrahedron would still work.) However, a minimum possible length would still be extremely restrictive, even including special right triangle arrangements. It only allows us to add these five possible arrangements (I'm not counting congruent figures as unique arrangments) to our short list:

#1

o


o---o

#2

o---o


o---o

#3

----o


o---o---o

#4

o


o---o


o

#5

----o


o---o---o


----o

Even with these additional arrangements, a universe in which there is a minumum possible length can contain no more than five particles, in only eight possible configurations. As you know, the universe contains many more particles than that, and in many more arrangements. So there is still a severe discrepancy between the logical consequences of a minimum possible length and our observations of reality.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">How do you even locate two points in space if there is no smallest dimension?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

We arbitrarily select a length, usually based on some physical quantity or process (for example, the meter is defined in terms of the speed of light), to be used as a stardard unit, and then note how many of those standard units must be combined to connect one point to another.
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

As you see, even in an infinitely divisible universe, we have to for practical reasons, divide it into manageable parts. I am not following your argument why there can only be 5 particles in the universe given a minimum length exist. Do you remember playing with ball and stick models in chemistry? Let's say I were to give you a very large set of balls and sticks but all of equal sizing, and I gave you a couple of simple rules for constructing 3 dimensional structures using those balls and sticks. 1. Before any two balls can be connected, there has to be a stick between them. 2. You are allowed to glue the sticks (or rather, the ends of the sticks stick to each other) to form longer connections between balls.

The smaller the structures you construct, the less possible arrangements are possible. But when you start making larger structures, all kinds of shapes and angles become possible. I find it easier to think of it this way than trying to deduce the structures through mathematics- which might take forever.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by jrich</i>
<br />1234567890,

Does your finite divisible universe also require that matter have some smallest size? Is there a smallest particle?


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

Yes there is since we assumed there to be. I really can't see how one can deduce through logic whether the universe is infinitely divisible or not. Those seem to be two mutually exclusive axioms.

If you start out with an infinitely divisible length, how can you have a minium length? Conversely, if you start with the assumption that 1 is the smallest non-zero quantity, how can you divide it into halves without contradicting your starting assumption? What is 1/1?

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by rousejohnny</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 jrich</i>
<br />1234567890,

Does your finite divisible universe also require that matter have some smallest size? Is there a smallest particle?


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

I have this question too, if there is a smallest particle the divisibility would have to be in interger multiples, or?

I like Jan's wave idea and thinking in terms of plasma. There would be an infinity of divisibility in the plasma (an aether) when homogenious and without dynamics. But, in order for dynamics to occur there would have to be defined multiples for interaction and complexity. There plasma would have to have or acheive differential charge as well in order for dynamics to occur.

So maybe both assumptions are correct, depending on the context of the debate.

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

Well, let's assume for a minute that the smallest particles are the electrons and positrons. If they are the smallest, then you can't divide them any smaller. But if you were to arrange them in space,
the resulting structures would be a composition of individual smallest particles and not fractional pieces of it.

Of course if you started out with the assumption that something is infinitely divisible, then it is. This however is not deducing infinite divisibility.

And as you, I think both assumptions are valid, but one excludes the other.

<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>
Yes there is since we assumed there to be. I really can't see how one can deduce through logic whether the universe is infinitely divisible or not. Those seem to be two mutually exclusive axioms.

You start with any finite quantity, how do you divide it into halves unless you assumed it to be divisible first? Likewise, if you start with the assumption that 1 is the smallest non-zero quantity, how can you divide it into halves without contradicting your starting assumption? What is 1/1?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
You are correct that one cannot apply operations which contradict the assumption as that would be a form of logical fallacy. However, this does not mean that one cannot show that the assumption leads to logical or empirical contradictions. I provided such a proof in the previous topic on this subject last week. That was why I asked the question, to see if this ground had already been covered.

If assuming both finite or infinite divisibility in turn leads to contradictions, then it is a matter of personal preference since there is no way to scientifically distinguish them. However, when one assumption leads to contradictions and the other does not, then no matter our personal prejudices we must be accept the correctness of the logically consistent one. If one does not accept this then one is not practicing scientific, critical judgement and will not be taken seriously by most of the participants in this MB.


JR

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jan</i>
<br />Skarp,

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">I don't see the need for a medium<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

I stumbled a little bit over the wave issue and it just didn't come out right. What I wanted to say is that waves without a medium do not make much sense to me. I still see a wave as the collective motion of objects. But if <i>everything</i> is wave-like, then this would imply that there is no such thing as a smallest particle. For let the smallest particle be wave-like, then this particle must be the collective motion of other particles. Hence, a contradiction with the smallest particle assumption. There can be no smallest particle. []
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

An ocean wave, overly simplified, is the collective motion of H20 molecules, which themselves are just an arrangement of units of "particles" we call protons, electrons and neutrons. So here we have a wave that is made up of indivisible units, if we assumed for a minute that the electron, proton and neutron are indivisible units. I mean we can actually "count" the number of these units that comprises a particular ocean wave frozen in space and time.

Does wave motion lead to infinite divsibility then? Only if you first defined a wave as an indivisibile "particle" then tried to show that by another definition of a wave, that waves are the collective motion of many particles, waves are not indivisible. I think all you have shown using this method is that both definitions have mutually exclusive meanings. It is not a logical deduction.

Well, perhaps you have also shown that the idea of matter-waves is quite vague in QM.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by jrich</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 1234567890</i>
Yes there is since we assumed there to be. I really can't see how one can deduce through logic whether the universe is infinitely divisible or not. Those seem to be two mutually exclusive axioms.

You start with any finite quantity, how do you divide it into halves unless you assumed it to be divisible first? Likewise, if you start with the assumption that 1 is the smallest non-zero quantity, how can you divide it into halves without contradicting your starting assumption? What is 1/1?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
You are correct that one cannot apply operations which contradict the assumption as that would be a form of logical fallacy. However, this does not mean that one cannot show that the assumption leads to logical or empirical contradictions. I provided such a proof in the previous topic on this subject last week. That was why I asked the question, to see if this ground had already been covered.

If assuming both finite or infinite divisibility in turn leads to contradictions, then it is a matter of personal preference since there is no way to scientifically distinguish them. However, when one assumption leads to contradictions and the other does not, then no matter our personal prejudices we must be accept the correctness of the logically consistent one. If one does not accept this then one is not practicing scientific, critical judgement and will not be taken seriously by most of the participants in this MB.


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

I assume you are referring to your argument regarding to touching spheres where the intermediate space directly above and below the
point of contact would be smaller than the diameter of each individual sphere? One obvious answer would be that at that distance, there are no spheres. Again, all the arguments leading to infinite divisibility appear to arise from a particular geometry , which carries with it a set of assumptions. If you start out with different assumptions, you get different results. And you can't say one is self contradictory unless you interpreted it using a different set of assumptions. It's like you can't say the shortest distance between two points is not a straight line leads to contradiction unless you used Euclidean geometry.