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IDEAS, MATH AND LOGIC
Tony Giovia
www.geometryofideas.com
> Since the joining of particles is governed by forces measurable by mathematics, and insofar as mathematics is a subset of logic (or vice-versa ! ), then this joining is what we call logical thinking.<
Detail for me, if you would, how you mean "logical" in this context. I take it you don't intend it's most common meaning (linear, empirical) but rather some organizing principle vs. something chaotic? Or?
Hi Susan,
Me >>> Since the joining of particles is governed by forces measurable by mathematics, and insofar as mathematics is a subset of logic (or vice-versa ! ), then this joining is what we call logical thinking.<<<
You >>> Detail for me, if you would, how you mean "logical" in this context. I take it you don't intend it's most common meaning (linear, empirical) but rather some organizing principle vs. something chaotic? Or? <<<
An empirical, dimensional organizing principle - creating a stable physical structure that, like any physical structure, requires a force to change its design.
The image you quoted above is an engineer's definition of a logical thought process. The entities of the picture can be any context you choose - individual elementary particles, small groupings of particles into individual ideas (the grail of "Idea Elements"), or complex systems. Since these entities are physical objects, the relationships among them are mandated by the laws of physics - which determines the design that the elements form and the strength of the structure.
We understand physical laws in terms of mathematics, and we understand mathematics in terms of logic. Again, an engineer's definition, brought to you by Merriam-Webster :
Logic - "The fundamental principles and the connection of circuit elements for arithmetical computation in a computer."
There is a distinction here that I hope I have made clear. I am assigning idea values to the relationships among particles, not to the particles themselves. Ideas = Relationships. This is not arbitrary; it is because any material object can be broken down into smaller objects, so it both contains relationships and exists in relation to other external objects. (An "object" or "entity" is anything that can be addressed - "sensed" - by a mind.)
Tony
Phil, please consider this response as a counterpoint to the idea you have presented, an idea which as you know has been accepted scientific fact for many years.
>>> (3. Stable Laws) All the "laws" we know of are only approximate, are probabilistic [quantum mechanical] in nature and have been tested in only a tiny portion of the known volume of space- time. <<<
This is a common objection, but one which I am not convinced falsifies the argument that math and logic can reliably understand part or all of the universe. I think the reason this objection has the influence it does is that there has never been an accomodation for the theory that ideas have a physical existence.
Certain laws are not approximate: 1+1=2, 1-1=0. As you know, these are the only operations required to construct all the logic gates used in a computer. Which are all the operations required for thinking and creating a consciousness in a machine (see THOUSAND WORDS <g>). To me this says that a mind or machine composed of these laws is not a probalistic entity.
Put another way, since laws are creations (ie copies/interpretations of sensory inputs) of a mind, and the simple laws that operate a mind are not probalistic, then these creations in their physical existence are not probalistic, or at least not necessarily probalistic. And that is the point - math/logic creations are physical entities, objects within objects, that exist and "work" in their own contexts. ( I think this gives new meaning to the word "psychotic" - hey, no wisecracks!)
The math/logic operations which compose our brains are (obviously) the very ops we use to think. It is not possible to trust the thoughts in our own minds, the constructs of our own minds, if we don't believe that math and logic are descriptive of some part of reality. Which begs the question - How is it mechanically possible for a mind to reliably doubt the stability of the very math/logic ops that compose it? It may be Godelian, but it is not probalistic.
It doesn't seem arbitrary to me to reason forward from a unique point (the Big Bang) and conclude that every single thing created by that point is somehow related to its birth and to all its brothers, sisters and descendents. If anything, this context is large enough to include even probalistic events. And if every single thing is related, then what we call "probability" can also be called "lack of knowledge". By "related" I mean math/logically (mentally) associated, including the fact that "probability" is itself a mathematical concept.
So worst case you have a mind functioning with non-probalistic math/logic ops observing (ie "mirroring" or "copying") a so far partly-probalistic universe. I say partly-probalistic because at the least the mind observing the universe is also a part of it.
To sum up:
1) There are probalistic elements in the universe.
2) There are non-probalistic (math/logic - mental) elements in the universe.
3) Both probalistic and non-probalistic elements are physical entities,
and "work" within their contexts.
4) A non-probalistic mind is used to investigate probalistic events.
The point I would make is that quantum mechanics is no more or less a complete description of reality than Newtonian physics or Euclid's plane geometry. It is always a matter of context. And of course the mind that says quantum mechanics is probalistic is using math/logic ops to make that that determination; meaning you have to take a stand on the stability of mental operations.
Is reasoning forward from the Big Bang a fallacy? I can't answer yes to that without questioning the efficacy of the very logic that composes my thoughts. Will parts of that reasoning reflect reality? Of course. Will all of that reasoning reflect all of reality? Can't say; maybe it will only explain one of the layers, like plane geometry.
Are you right? Sure, as far as it goes. Am I right? Sure, as far as it goes. It is all a matter of context, of the filters you bring to the problem. Unless I am missing something ( mwa? ) I don't see the conflict. Until it is proven that math/logic ops are probalistic events, I say keep the definitions exact and the assumptions supportable and go build your world.
Bottom line: Anyone agreeing with the point of view presented here must at the least agree that ideas can be viewed as physical objects structured from non-approximate, non-probalistic laws. If that filter is uncomfortable, then we just have to agree to disagree. Like a couple of bluebloods. <g>
Tony
Phil,
Maybe my argument wasn't so definitive! But for the record:
>>> I don't call things like the above "laws".. they are logical consequences of accepting Peano's postulates. <<<
We may actually be at a point where we cannot agree on this. Miriam-Webster defines a law as "a rule or principle of construction or procedure." Minds are constructed of AND and NOT gates which combine to form the other logic gates. These gates are "rules or principles of construction or procedure." These gates are the embodiment of the concept of "laws " (as defined by MW), actual physical definitions.
Two points:
1) Netscape kept crashing on me when I try to access info on Peano (I don't remember the axioms), but even without them in front of me I can say that since you accept Peano's axioms as assumptions, any conclusions you draw from them are built from AND and NOT gates, which I define above as laws. After all, what do the words "logical consequences" mean if not that something flows from another by reliable and lawful means?
2) When a computer solves a F=MA problem it is using AND and NOT gates in combinations to do the work, meaning it both understands the equation and solves the problem using primitive non-probalistic steps. And F=MA is also a "rule of construction".
>> And, if you accept ANY system that contains them you can construct formally undecidable propositions within that system. That's in "Godel speak". The same thing in "Turing speak" is that you can always find statements within such systems that would require an infinite number of steps to prove. <<<
But these are not probability problems, they are design problems. Or at least let me say that I do not see how probability fits into this observation. My contention, remember, is that logical elements in the universe exist independently of the probalistic elements. Logic is not structured with probalistic elements, although it can catalog probalistic events - two different things.
What drives me is the seemingly irrefutable logic that ideas were created by (or are a result of) the Big Bang; before the BB there was nothing, after the BB there was enegry and matter and ideas. Stating that ideas exist independently of energy and matter, but that ideas can still in some way influence energy and matter, is to me insupportable (if that is what science is saying today). I am much more comfortable trying to defend the "ideas as energy and matter" thesis than any "ideas are an undiscovered form of existence which affect matter without, er, actually touching matter" theory.
>>> I like to think that most of what I write can be traced back to pretty solid roots .... <<<
I know, that's why I like to talk to you and Richard. I'm not exactly out there chasing butterflies either.
>>> Like you, I am uneasy that we have no F=MA or E= mc^2 for ideas. However, I am inclined to think of ideas, as being independent of the substrate upon which they are exhibited. <<<
They have an independent existence, but they are not composed of anything other than energy, and they obey the same laws that energy obeys - unless and until somebody discovers a new form of existence. Sir. <g>
Tony
Richard,
>>> 2) Scientific American, "Quantum Mechanical Computers", pg 142, 10-95:
"Logic gates are devices that perform elementary operations on bits of information. The Irish logician George Boole showed in the 19th century that any complex logical or arithmetic task could be accomplished using combinations of three simple operations: NOT, COPY and AND." <<<
Apparently NOT, COPY and AND are the three basic building blocks relating logic and mathematics. My logic book was no help in defining these terms, so I turned to my computer books. There I could find no reference to a "COPY" gate ( the terminology is new to me). SA offers the following standard circuit notation for COPY:
--------------- 1
/
1 ---------- /
/
--------------- 1
Based on this I offer the following definitions:
AND = Join *perceptually independent* data structures (sets) directly together.
NOT = Do not join *perceptually independent* data structures (sets) directly together.
COPY = Duplicate a *perceptually independent* data structure (set).
>>> Furthermore, many of our "set in stone" beliefs about physics are quite rightly held without the possibility of any sort of repeatable experiment. The big bang is a rather noteworthy example. Granted, small scale empirical investigation implies that it did happen.... <<<
There are two points here: 1) The quality of the evidence that the Big Bang occurred, and 2) what if the Big Bang theory is disproved?
As to 1), even you agree that there is physical evidence that such an event occurred. In other words, people on both sides of the issue agree that certain empirical evidence exists. Therefore interpretations of this evidence, though they may diverge, must include this evidence as a basis for reaching their conclusions. That's ok and it is done all the time. But the point is that if everyone starts with the same physical evidence, and everyone applies the same logical/math operations to the evidence in the same order, then everyone will perforce reach the same conclusion.
Three ways conclusions can diverge is if 1) the math/logic ops are applied in a different order to different pieces of evidence ( as in x + y - z and x - y + z ), 2) some people do not use the entire set of available evidence, or 3) assumptions not in evidence are included in the calculations. This last is not necessarily a negative (as far as falsifiability is concerned) as long as it is also based on verifiable evidence.
As to 2), if the theory is disproved it will be because the replacing theory found a better way to organize new and existing evidence - the facts that everyone agrees on today will not disappear when a new theory is adopted - that evidence will still exist, and will still be included in the new theory. Again, falsifiable vs non-falsifiable.
>>> ... it is naive to distinguish the spiritual from the scientific on the basis of either a verifiability theory of scientific investigation, nor on one based on falsifiability. <<<
I think I have answered this (in my own naive way) in the preceding paragraphs. Fifty people starting from fifty different starting points (that none can agree on) will most likely produce fifty different conclusions.
I think our differences on this point come down to a single issue - I see no distinction between "opinions" and "non-verifiable conclusions". If you could explain the difference perhaps I could better understand your viewpoint.
Tony
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