Students everywhere are routinely taught how to arrange pieces of
educators who are unaware of elementary pieces of information that
to be easily arranged. I see that as a huge disaster. When they
do become aware, I expect
they will come to realize that the teaching of mathematics and
other technical subjects is
best redeveloped using well-designed information building blocks
That approach to teaching may also become a favored way of
foundations of mathematics.
If you think that "what is needed today is a new foundation for
mathematics" you may
want to look into that possibility further. If you do, I would be
interested in your reactions.
There is explanation in the first 2 papers on my web site.
On 3/3/2015 7:39 PM, Bruce Schuman wrote:
Bruce: I wonder why we presume that a "theory of everything" has to be grounded in physics.
John: The idea of a GUT was first conceived by the physicists, whose theories are more precise than theories in other empirical sciences. . . .
Versions of calculus are used in all theories of physics . . . Bringing physics into the discussion seems to be more confusing than enlightening . . .
Bruce: I am still hung up in this issue of abstract symbolic representation of "real world" processes and objects.
For me, this gets right to the core of how language -- and maybe (?) calculus -- describes reality -- following the precept I absorbed so many years ago, and which does seem foundational in its simplicity: "reality is continuous, concepts are discrete." (does that include calculus?) When I first got into studying the mathematical structure of language, I was looking at "modeling languages" and trying to understand the connection between an abstract symbol or "state in a medium" and some object "in the real world".
Language is composed of "little boxes" with names/labels that define categories supposedly bounded by known dimensions -- "how we can tell the difference between a dog and a wolf" -- and what those dimensions are is generally a matter of social stipulation or convention. Something is inside that n-dimensional box -- whether it is a wolf or a dog -- but in either case, we are talking about an abstract symbolic model -- existing in some medium such as a computer -- which we then map to some real world object ("that animal over there").
I remember taking my first course in calculus, and my very smart and respected teacher Miss Johnson trying to explain the concept of a "limit" -- and admitting to everyone in the class that she did not understand it very well. In my case, sadly, it's still true. I have never really grasped the notion of a "limit" as it is used -- so successfully -- in calculus.
There is this thing about "rational numbers" -- which maybe I abuse by wanting them to be represented in a finite number of decimal places -- because "that's the reality of what we can actually do". So, the way I see it, any number in a finite number of decimal places can be represented in a hierarchical/taxonomic grid where each further level of decimal place is like the next lowest level of "taxon" -- with continuity -- the unreachable, the infinitesimal -- at the bottom of the cascade.
So -- the difference between a wolf and a dog is a matter of degree (in any number of dimensions) and boundary-value conditions -- that we sneak up on -- like a limit -- in tiny increments, until some "tipping point" is reached.
This line of reasoning (if it IS "reasoning") -- tends to convince me that this GUT ambition when grounded in physics is eternally vulnerable to the confusion John mentioned in his first comment in this thread:
"Physicists are the closest to finding a Grand Unified Theory (GUT) of everything. But every time they find one, it opens up far more mysteries than it solves."
This is why I am inclined to look for a general solution in "pure mathematics" -- and probably not "complex" mathematics -- but instead, a kind of mathematics that is so simple it's almost inconceivable -- "simpler than we can conceive" -- the kind of definition that might emerge from a famous question like "why is there something rather than nothing?"
I do tend to be somewhat guided by holistic guesswork and am perhaps vulnerable to its fantasies -- but I like the title of a book by the Dalai Lama -- "The Universe in a Single Atom -- the Convergence of Science and Spirituality". The Dalai Lama is a very well-educated guy -- who has spent his life travelling all over the world going to conferences and talking to very smart people, including many leading scientists.
So -- I am inclined to look for a mathematical/conceptual foundation where something dimensionally confusing happens -- such as "the infinite plugs directly into the infinitesimal" -- the infinitely large maps directly to the infinite small -- somehow across descending scales of "self-similar" ("fractal") recursion -- with "everything" that can be conceptualized or mapped to or described by numbers bounded and described within that framework. As I see it, natural languages -- and machine languages -- are parsings within that general framework -- of "all possible conceptual form".
So -- what does this have to do with physics? This might be wrong-headed, but if physics insists that reality is grounded in ever-smaller particles -- and every particle has to be bounded in some way that makes it distinct -- in the end, we are talking about the problem of mapping a complex body or cascade of interconnected distinctions to some "object in the real world" that supposedly has these properties. Can we EVER do this without measurement or round-off error -- ? No -- I would say -- because "reality is continuous and concepts are discrete."
I tend to want a simple framework defined in pure finite-state math that maps directly one-to-one to a machine state, where everything in reality is contained and categorized by dimensional boundary values in a finite number of decimal places. The "real number line" is probably the bottom level of this cascade -- as an unreachable/unknowable limit -- that is defined as "unbounded" in some mysterious way we have not quite conceived.
And at the top of the cascade -- the taxonomic hierarchy of numeric decomposition -- is the un-namable unbounded undivided infinite unit "one" ("the whole", "the all", "the Tao that cannot be named") also somehow magically self-similar to every other unit in the cascade all the way back down ("turtles all the way down") to the infinitesimal as a limit at the bottom.
If we could figure out the top and the bottom of this form, and show how it maps to itself ("the universe in a single atom"), we might (?) generalize the principle of conceptual form, whether "bottom-up" and observably mapped to real experience, or "top-down" and purely transcendental and theoretical -- with "everything else in between".
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F Sowa
Sent: Tuesday, March 03, 2015 11:17 AM
Subject: Re: [ontolog-forum] Grand Unified Theories
David and Bruce,
Johnson at a Management Accounting seminar in 1992 ...
"We have come to the end of our ability to understand & describe the
world around us within the confines of Newtonian physics."
Johnson was about a hundred years late. In the 1890s, physicists began to see that Newtonian mechanics had serious limitations.
In 1905, the "annus mirabilis", Einstein pointed the way to the future with four amazing publications:
1. Brownian motion: He showed that particles that are visible
in an ordinary microscope are being kicked around by much
smaller invisible particles. This was the first direct
evidence for the atomic hypothesis.
2. Quantum mechanics: He showed that Planck's theory of radiation
implied that light energy is quantized in discrete photons.
3. Relativity: Physicists knew that Maxwell's equations for
electromagnetism -- which predicted that the speed of light
is the maximum possible speed -- are inconsistent with
Newtonian mechanics. Einstein developed a new theory of
mechanics that is consistent with electromagnetism.
4. Mass-energy equivalence: The famous E=mc^2 and its derivation
from the theory in paper #3.
For a summary, http://en.wikipedia.org/wiki/Annus_Mirabilis_papers
Double entry accounting—to which we are all beholden—was created from
the same intellectual ferment that produced The Calculus.
Versions of calculus are used in all theories of physics. I don't know what metaphor Johnson was using. But bringing physics into the discussion seems to be more confusing than enlightening.
I wonder why we presume that a "theory of everything" has to be
grounded in physics.
The idea of a GUT was first conceived by the physicists, whose theories are more precise than theories in other empirical sciences.
Researchers in the other sciences tend to suffer from "physics envy".
They compensate by trying to be "more mathematical than thou".
As Marcus and Davis pointed out, they tend to "decorate" their theories with abstruse mathematical equations that are more confusing than helpful. Unfortunately, reviewers and investors who are hopelessly confused by the math, don't want to admit their confusion.
ideas rule the world. We need to understand ideas -- good ones, bad
ones, confused ones, crazy ones. It's ideas that cause teenage girls
to join ISIS.
I agree. The ISIS propagandists wrap their crazy ideas in a thin veneer of theology. Marcus and Davis were trying to warn investors that a thin veneer of mathematics can be just as confusing.
I wonder why we presume that a "theory of everything" has to be grounded in physics. What about the approach that every "thing" is actually only knowable as a symbolic representation as some kind of concept -- including fundamental particles? Seen this way -- a "theory of everything" involves a "universal theory of concepts" and the issue becomes reliably connecting the abstract symbolic representation ("the concept") to the "actual empirical object" -- like a fundamental particle -- or any other discernible and bounded/distinct object.
I like physicist Arthur Eddington's suggestion in his "Fundamental Theory" (1946): "All the laws of nature that are usually classed as fundamental can be foreseen wholly from epistemological considerations."
My own feeling is -- what is needed today is a new foundation for mathematics itself -- a new ontology of mathematical fundamentals -- that interpret in mathematical terms what Eddington was calling "epistemological"
I like an approach that says something like "all concepts can be constructed out of dimensions and the fundamental constructive element is a cut (as per Dedekind) or distinction." You can build any concept taking that approach and it is 100% linearly recursive and extremely simple.
For me, all of this floats in a framework defined by John Sowa's basic proposition, as I read it: "concepts are discrete, reality is continuous". We need to generalize the framework that contains this fundamental truth, and show how all mathematics -- arithmetic, real numbers, continuous variation -- emerges as attributes of this containing framework. Distinctions within this framework become "all the concepts in reality".
Do this in a top-down way -- and you get something like the model described in the Numenta white paper on hierarchical memory, which mostly describes simple linear recursion. Build it up from the bottom (based on observable empiricism) -- and you get something like John Sowa's "semantic networks".
I don't know if "neurons" are organized in the way Numenta suggests -- but I'd say concepts can be -- and neurons support conceptual processing.
We should connect these approaches. We need a "transcendental container" that can hold all of this in one framework. This mathematical object might be fairly simple -- if dimensionally a little tricky -- and emerge as the conceptual foundations for a new and very-much simplified and integrated way to understand reality.
PS -- ideas rule the world. We need to understand ideas -- good ones, bad ones, confused ones, crazy ones. It's ideas that cause teenage girls to join ISIS.
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