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http://www.rbjones.com/rbjpub/logic/log001.htm



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Logic and causality are intimately related. If we know that A causes B then we can also say that A logically implies B. Thus, A=>B. However, consider the following logical equivalence:


(A=>B)<=>(~B=>~A)


If A causes B then A=>B, but this is equivalent to saying ~B=>~A. Is this then the same as saying ~B causes ~A?

What makes this so tricky is that in order for A to cause B, A must be an event that precedes B in time, but is it meaningful to say that ~B precedes ~A in time? If not, is it still valid to say that ~B *causes* ~A since A causes B?



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Because?
Because of the relationship between time and causation: for X to cause Y, Y must precede Y in time. That's in the nature of causation.


QUOTE
It would also be useful if you would not use strange symbols that disappear from the text. It makes it really hard to follow.
I take it that you don't see the logical connectives. Well, I'll take the blame for that. A certain countably infinite member complained bitterly that I didn't use the standard symbols. My reason was that I believed that some people did not have their computers set up in such a way that they could see special symbols, but because of my subsequent interaction with him (indeed, pretty much everybody), I was of the belief that these days, anybody can read the characters we put in our posts. But apparently my original belief was correct. I wonder how widespread this problem is.


QUOTE
Also, it was not irrelevant because it is certainly true that "if ~B then ~A" follows from "if A then B." The question then is, how do we resolve the apparent discrepancy between the two? In other words why is the TRUTH of ~A dependent on the truth of ~B if there is no causal relationship?
You're confusion a particular program regarding natural language, formal logic, and philosophy. Let me assume that you're read the threads I referenced, then as you will no doubt appreciate from that discussion, it is false that "if ~B then ~A" follows from "if A then B". This is a classic case of a floating abstraction. What do you mean by "if A then B"?

Let me take this opportunity to point out that man is not born with a priori knowledge of the laws of formal logic which are taught in typical undergraduate courses in symbolic logic. Formal logics are the highly developed product of intensive reasoning (and unreasoning). Your false presumption that ^B=>^A "follows" from "A=>B" is a case in point. Natural languages and human reasoning does not work on the basis of meaningless symbols like "A", "B", but symbolic logic does not work on the basic of "if" and "then". You should pick context: are you asking a question about the meaning of "if" and "then", or about symbolic systems?




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--Last edited by saucer on 2007-01-04 00:25:36 --

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QUOTE:

First, I want to ask you what Objectivism has to say about Mathematics, and if anyone in the field of Objectivism is working on advanced mathematics. Second, what do you think of http://www.wolframscience.com/ which isn't really science but a very narrow scope of mathematics? Third... do you think you have what it takes to tackle mathematics? Because that's what we're going to be doing in this thread.

I just realized that a theory of basic constructs would account for both mathematics at the most general levle and physics at the most fundamental level. This would not be a scientific theory, but would give rise to a whole range of mathematical possibilities, and define it well enough for us to be able to determine it once we see the world as it is.

First, for the science part, we need to break up the General Theory of Relativity and Quantum Field Theory into their mathematical components, and remove classical assumptions about the world. For the math part, we need to pretty much rely on logic, then widen its scope so that quantities and qualities are both dealt with. Then we merge the two. This is my strategy.

However, we also need to start at the basics. Why does 1 + 1 = 2? Because it is defined that way? Does 2 have an identity that is seperate from merely the additon of 1 + 1? If it does, then what is the definition of +? What is the defintion of 1? Can we define 1 as a multiplicative identity without defining multiplication? Does multiplication depend on addition? What is unique about 1?

I think that asking about the number 1 would be without context. 1 in what mathematical system? What are the axioms? In the real world, we'd be asking 1 what? In math, we have to ask what our axioms are. We cannot just have a naked 1. 1 must represent something. We may have to start with something more abstract, such as A.

Now as for +, that is a bit harder. We can ask the same questions, but we have to define it as an operator. Let's call it a. Notice how I use letters for both the number and the operator, but the operator is lowercase whereas the number is upper case. We will assume for now that numbers cannot act on eachother directly, but through operators. In later sessions we may find out that this is false, or that it's true, but we can interchange numbers and operators in the same way we can interchange dots and lines in geometry to get the same relations.

Now, = implies that if B = C then C = B. Otherwise, we'd use > and < signs. The basic way we do math is either AbC = D. Or aBc = d, but we can ignore the latter for now... but we shouldn't ignore the possibility in the future. Now we have to define context. We could do it this way: (AbC=D)_"1" where 1 is axiom 1. The _ designates that AbC=D operates under an axiom. Is this too complicated already? Or are you following me?

At this point I really don't know how to proceed without biasing myself. However, you see how I approach the subject, by using abstractions first then applying meaning to the abstractions based on context. Does anybody get it or am I being too vague? I'm basically using strings (not string theory type strings, but computer theory type strings) for lack of a better word. Would using propositional logic be easier for people or is this, what I'm doing now, a good idea?






DIFFERENT WRITER REPLY QUOTE;



Yes, although it is true that there is a method of reducing "2" to "1" and "+". For "13" (as an example), that number is a higher-order abstraction, also almost certainly "9". The numbers 1-3 and perhaps up to 5 would be fairly low-order concepts. This is an issue for psychology, and is related to the "crow" problem.

The root of mathematics is the notion of "measurement", and is important in concept formation (since concept-formation is measurement-omission). "1" is what you have when you unify "1 dog", "1 frog", "1 sock", "1 clock" and omit the measurement "which thing". You can form the concept "2" the same way. Part of forming these concepts is having the ability to distinguish, for example you have to be able to tell that "1 dog" is not the same as "2 dogs". That's not hard to do (at all). OTOH is it perceptually almost impossible to directly distinguish a thousand dogs from a thousand and one dogs, in which case you have to rely on a method (of "counting"). By that method, you can derive a massively broad range of numbers.  

--Last edited by saucer on 2007-01-04 00:41:00 --

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QUOTE:


i think maybe when you started this thread you were contemplating the same things as Rene Descartes was when he came up with his famous one liner, "I think therefore I am."  by this, descartes meant to illustrate that if we can all agree to the fact that humans think, which is irrefutable, then we can also agree that at least that much exists.  

you figure it out from there.




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http://www.blupete.com/Literature/Essays/BluePete/Argument.htm




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http://www.w3.org/DesignIssues/Logic.html


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Saucer

As one of the simple folk in regards to mathematics, i find greater comfort in either talking about the make-up of the dog. or what a hundred dogs creates. I wonder if that sounds logical.

   Once we begin to factor "whole sums" shown as abreviated signs , then I part company. The reason is my need to first trust the represented sum , by which i need the logic of someone who fully understands it . These signs can be created from pages of mathematics and many of them can be used in one line.
   i need a dog to have particular behaviours which I have by my own experience seen over a lifetime of owning a dog. Then the statement can be simplified to "dogs eat meat" or "dogs are territorial". Because I don't need the concept of territorialism explained to me.
   What often happens is the speaker uses part "dog" and adds (from thier own experience) other behaviours that cannot be part of the experience of owning a dog , but may be experienced if one were a vet.
  This deminishes nothing from either of our love of dogs. nor does it mean that we understand less or more about MY dog. Since I have spent lots more time with MY dog , I understand It's personality while You can tell me reasons for behaviours.
  If I told you nothing about my dog , but asked you to treat it over the phone , you could not. This is the same as me treating my dog without vet knowledge.
  So I guess I try to tell you guys about my particular "Dog" Which is dear to me, without having a full knowledge of medicine. You on the other hand , tend to answer me without asking about my dog.
So for me , constants are like all dogs being exactly the same ,despite having totally different backgrounds. (because any particle which has the same background came from EXACTLY the same source at EXACTLY the same time).
  your 1 dog is a simplification of a greater statement. The constant is required so I don't have to understand the greater sum. The rule (for me) breaks down when constants become absolutes. Like religion the stance relies on being totally correct at every prior point in the chain.

  I cannot see that history allows that to ever be the case.

Cheers
Iseason

I see your drift, (a little) Iseas.


YOUR QUOTE: ..(because any particle which has the same background came from EXACTLY the same source at EXACTLY the same time).
 your 1 dog is a simplification of a greater statement. The constant is required so I don't have to understand the greater sum. The rule (for me) breaks down when constants become absolutes. Like religion the stance relies on being totally correct at every prior point in the chain.

 I cannot see that history allows that to ever be the case... END YOUR QUOTE.




i, agree Iseas. one dose n't necessarily have to understand the "greater sum". ( it is NOT a contextual entree! per se.)


i, don't see why you see the "rule" breaking down. a "constant" is a notion, (and so..) an " absolute"! (the constants' definition: a constant is a fixed, but possibly unspecified, value). This is in contrast to a variable, which is not fixed.

that may seem even more "math & jibberish" ...BUT it is stating it's fact pointedly, and discerning a difference, no.


very very small tic

Very,very small tic

I understand what you stated. But that creates a very big "theory of chance".For constants to be "absolutely correct" , the universe must have 'cloned ' each particle. Otherwise , there must be variation. I see no other level which we speak about where we would expect any two things to be exactly the same.
  We may not be able to measure the difference but no two points in space or reality can be equal.This is impossible. Yet we accept that the similarities in constants are great enough to base major science on this being true.For most applications , I agree this is deep enough. For most things that we can do with particles , it is very new ground. But when you want to understand the origin of "one" particular event, The variation becomes huge unless they "are the same".

Cheers
Iseason

Quote :

Iseason wrote : Very,very small tic I understand what you stated. But that creates a very big "theory of chance".For constants to be "absolutely correct" , the universe must have 'cloned ' each particle. Cheers Iseason

I think I concur with you Iseason;


IN the area of this "cloning" (..ie; of a Particle!..) - THOUGH THE 'ORDER' OF ANY CREATION MAY BE: STUPIFYING! AND ..  "FUNDAMENTALLY" "TRANSCENDENT!" IN PERPETUATION ...

...IF we as Humans are to grapple with the idea "rightly" it pre-requests' "us" to believe: 'constants' ARE "absolutive!"...(incondite) definition.



ferme  

--Last edited by ferme on 2007-04-02 19:15:04 --

I agree Ferme.
In order to grapple with theory , there is a limit to what we can call "absolute".
 But at the moment , anything that ignores the absolute fundamentals in order to see further is ridiculed.meanwhile , solid theory is based on the assumption that fundamentals exist at the level we percieve. This is much more damaging to finding the "reality" that the fundamental rules are to make.
 without having a starting point which is unassailable, There will always be room for a "pre-theory" . that is why I have gone to where I see the beginning. not to fit my theory , but to allow all posibility without leaving the gates wide open.
  you cannot divide a singularity. therefore the original state must be so. In order to progress , methodology means it must propagate itself.It cannot be in two places at once , but can be in two inbetween states at the opposing ends of reality. This means it can never meet itself as mass.

Cheers
Iseason

YES Ise,

I whole-hearted AGREE (with your last Message.) You put your idea, succinctly!





I do NOTthough, that: quote " You cannot divide a singularity" end-quote!

..THAT (it, seems to ME).. is quite "loaded." A real plateau[/couleur]!

[couleur=#000ef0]ferme.  

--Last edited by saucer on 2007-08-27 00:15:51 --

Hi Ferme

I guess you must agree with me on what I mean by a singularity.

As I have said, I could not begin to judge it's size, But whatever it is, it is indivisible.

To have it not be small. (say I instaed occupied the universe as a single mass) I would need to describe "area", because if the universe is the size it is and the particle occupies it as a mass, Then it needs "motion". This is against my model.

The motion That makes the particle be "everywhere at once" is quite seperate from the particle state. The energy of the particle does not reduce , nor can it be shared. But it can appear to be more often in one place than another.

This has nothing to do with the particle state but has a lot to do with where in the energy scale, the speed gives us the illusion of reality.

The change in states takes place once , at a constant speed and uniformity. However, Because we see both original and changed states, can revisit what occurred in every concievable order.the speed makes the difference.
That's what Einstien's theory shows.That if we travel at  higher speeds , the world slows down. Because we see through the hall of mirrors , and can only ever move at a sluggish pace , we are governed by the relative laws of that speed. Other realities are taking place(relatively speaking) and perhaps each life cycle takes us back through the same process at a greater (or slower) speed. Meaning , Forever and infinity are quite possible within this frame work.
Cheers
Iseason

quote:

the universe must have 'cloned ' each particle..


unquote.

Hi Iseas,

What about Fusion-physics and Giant-gaseous interstellar clouds of dust, hydrogen gas and plasma Nebulae, Iseas? Are those forged differently?  

--Last edited by zee on 2007-04-07 07:53:16 --

Hi Zee

I'm afraid this is a case where "one size fit's all". REALLY.

The fact that you are talking about 'actual behaviours or things' means the particle size is a long way shorter than your perceptions. otherwise constants would have noticable variations.

it's like us watching tv. Our eyes can't tell that the pixels are appearing 30,000 times a second. But a cat can.(perhaps we need to look through the eyes of a cat to see what is actually going on.)

there is no variation in the particle that I am talking about. just the "variation of occurances" .Because we can percieve both a past and a future , we need to be inhabiting a central point in the change. I just need to factor space into things.

It seems that the only thing every one seems to have solved since Newton's time is what is stumping me. Getting space to equal one at the same time as allowing motion.

Cheers
Iseason

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Maybe the Particl size we should be "relating to," is in: "Extra-dimension;" therefore its conceptivity MAY "..NOT BE VIEWED" IN "AN ORDINARY-WAY ...."


Maybe, "Our perceptions..." are, a; "Generic-splice.." "..just out there...." ..in Time ..! Like some crazy Metronome-clock. A "preset and a signature!" "just existing ......"



http://www.plastelina.net/games/game1.html



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--Last edited by saucer on 2007-08-27 00:12:51 --

differing dimensions is a definite probability,though they can only inhabit differing time zones and therefore differing particle sctructures and speeds.
the joining of the various dimensions create the whole fabric of space and so can never be seperated to allow one to be divided from the other. you cannot leave yours because you inhabit them all anyhow.
the only possibility is that outr technology allows us to understand more of each (big and small).

Relativity allows time to differ from viewpoint.An ant has a very different size scale to us. they notice things we don't because individually small things are commonploace.
We are entering that world as aliens because we cannot live it , only observe.

it would be possible to call an ants world as another dimension.We often think in terms of shrinking ourselves to ant size,just as we think of using a wormhole to go somewhere in space.
both may exsist but are illusive to us,but exist with us everyday,so dimensions are not something elswhere but inhabiting this reality differently.

Cheers
Iseason

You cannot leave this dimension ..in a physical-sense, though one must be (assuming: a definitveness in this respect.)

Ameny IV