you are a work of art (poem)

You are a work of art


I am Luna, reflector of light
I am the roiling, dark water below, glistening fluid
I am the pitch black clouds that seek
to bring chaos through blindness
I am the work of art of the Crafter
Who sees below all seas
I am a pawn on God's 17-dimensional chessboard
The 17!

the abstract contains the concrete, so it is said

"It really reminds me of the concepts abstract and concrete.  Why would concrete be any more real than abstract?  Because abstract can only be perceived by the mind?  Our brain is a perceptive organ, like the eyes and skin...Just that it perceives things about the mindscape which includes the abstract."

The importance of 10 and other musings

In the base ten system, 10 is ten.


In the base two system (binary), 10 is two.


In the hexadecimal (base 16) system, 10 is 16.


In fact, in every base b system, 10 is b.


In the base pi system, 10 is pi.  Note that 10 is an integer with respect to the base pi system.


So what makes an irrational number irrational?

I feel like I'm in a dark room with a few candles, one of which is math.  I have selected to focus on that candle but have so far found only very disturbing things by investigating this candle which I used to think illuminated the whole room; sadly, I don't think it does.  It is just another tool with its own set of applicability and limits.

If you're trying to understand reality in whole or in part, would you want to know "the truth" if it is so disturbing that it makes a significant percentage of people who "know" that "truth" have mental breakdowns?  In essence (though I loathe these terms) would you trade all of your sanity in order to learn this truth?  You might be set free by truth and maybe not but if you're insane (whatever the hell that means)but know the truth is that worth it?

Perhaps the process of becoming free, in mind at least, involves, or can involve, what some might call insanity.

Prococreators

Keeping in mind things like QM and some of its counterintuitive theories, what religion tells us, what mythology tells us, what spirituality tells us, what common sense tells us (and doesn't tell us), and the possibility of math being a creation of humans instead of discovered by humans, do we create reality?  A nightmarish scenario at times to be sure but then again nightmares are also dreams, the artwork of the subconscious, ghoulish as that might be.  Do we create reality?  Who created us, then?  The primal cause?  Maybe we are the primal cause if we create reality.  Who else can create reality and why hasn't someone who can create reality left us evidently alone?  Why hasn't such a someone destroyed reality, or at least all possible worlds with us in it?  I suppose if you can answer that, you might have an answer to the question "what is the purpose of humanity."  If we create reality, we are procreators, and if all (or most) humans create their reality, then reality is our prococreation.

We will never have a TOE in the Tegmarkian sense by a simple counting argument

This post is meant to argue against a Tegmarkian TOE which he calls a complete description of reality.  Towards finding a complete description of reality, one might just try a list of properties of this TOE such as its length and such as its descriptive power.  Are there zillions of concepts and experiences that we just haven't encountered so we have no words for them?  Could a zillion new words be added to the dictionary?

How about infinitely many new words?  Can reality ever be completely described if the dictionary is infinite?

Here is how I think of part of it: I believe that reality is infinite.  There are a few ways to prove that, depending on what we'd like to assume to make the argument go through.

I believe numbers exist in an abstract sense (i.e., for the sake of this argument, this is an axiom).  If you spend much time studying infinity, you at least get introduced to the smallest and next smallest transfinite numbers.  The smallest transfinite number is called Aleph Null or Aleph Zero and it is how big the set of all natural numbers is.  The next transfinite number is that of the set of real numbers (under the right side-axioms).  That means that in Hilbert's Hotel of infinite rooms, if each room had a natural number designation (or "address" (think computer memory)) AND if each room is filled, there is no way to accommodate the bus load of new patrons, each indexed by a real number.

The set of all natural numbers is said to be countably infinite.  The set of real numbers is said to be uncountable, as is every type of infinity beyond this.

The kicker is that, without going too much into formal systems, there are more real numbers than there are descriptions of all real numbers.  A description is a finite quantity but since there is no length limit inherent with descriptions, there are infinitely many different descriptions but only the countable type of infinity, that of the natural numbers.  Since there are more real numbers than that, (recall it is uncountable while the set of natural numbers is countably infinite), there are more real numbers than can ever be described.

If reality is at least as infinite as the set of real numbers, then we are done: we will never have a complete description of reality, no TOE.  It may be less obvious in the other cases: (1) If reality is finite then since you can add infinitely many neologisms to describe a finite reality, and most descriptions of reality won't even be needed, given that there are finitely many real things.  Finitely many things can be completely described.  (2) if reality is infinite, then it is possible that reality can be completely described and it is possible that reality cannot be completely described.  If the latter, we are done again since that would imply there is no TOE.  The trickiest case as I see it is what happens with when reality is infinite and can be completely described. Then a TOE would exist, although it may take eons to understand to even write it down, like Graham's number.

So I'm basically saying that a simple counting argument torpedos the notion of a usable TOE defined as a complete description of reality.  Sad to say it but that's where the investigation goes.  I still maintain hope that a TOE can not only exist but translated into more normal speech.

nth operation

n-th operation.  The successor function outputs a natural number that is one greater than the input (n=1).  Addition is repeated use of the successor function (n=2).  Multiplication is repeated addition (n=3).  Exponentiation is repeated multiplication (n=4).  Towers of powers are repeated exponentiation (n=5).  ??? are repeated towers of powers (n=6)?  ??? are repeated ??? (n=7)?

The number of atoms in the universe is estimated to be 10^80 atoms.  The required computation would involve just the level of exponentiation.  So it boggles the mind how the numbers involved get so big so fast.

Matheism is applying mathematical ideas to spiritual ideas and vice versa.

Cantor believed in an absolute infinity that could be the mathematical equivalent (think isomorphism) of God, what ever that might turn out to be.  Then Bertrand Russell came along with a cute way to prove that absolute infinity does not exist within Cantor's paradigm; Cantor is definitely  one of the most important matheists who ever lived.  He is the father of modern set theory.   He was one of the first to really delve deeply into how to mathematically formalize infinity.  Cantor was ahead of his time, imho, especially now that there can be different kinds of sets than what Cantor saw in which it is possible to mathematically formalize the concept of a largest infinity, an all inclusive set.  I believe strongly that Russell's proof that there is no absolute infinity consistent with Cantor's version of set theory (which is now highly prevalent in most areas in math) is what led to Cantor's psychic breakdown, something that happens all too often with mathematicians.

Cantor believed that sets can be thought of as collections.  So it appears that he thought of God as the collection of everything, an ensemble of all that is.  I am certainly contemplating whether God is merely a collection.  A collection of vibrations would, I think, comprise the aspect of reality corresponding to our material  universe.  That would automatically mean our material universe is contained by God.

The Pythagoreans also believed that math can be used to perfect the spiritual wealth and purity of the student.  Oddly (pun intended) enough, they thought various numbers had personalities.  For example, there are types of numbers called odd and others that are called irrational.  I believe the Pythagoreans were projecting their conception of humanity onto that which is pure, unfleshed and clean of human taint. In other words, I don't think they were exactly on the quickest path (which usually the steepest) towards greater understanding.

It is in matheism that sacred geometry, astrology, astronomy, Tarot and other forms of divination, numerology, etc., are all siblings from a common father, as it were.  These are intricate ways to store vast amounts of information and reveal how various types of things are connected to each other.  Matheism, it could be said, involves connecting the dots, as it were.

simply a test of whether I can or cannot export an LyX file to html

This is just a “proof of concept” check to see how well (and not so well) I can export from LyX to html to post on my blog.

Proof Schemes

Proof schemes are deduction or transformation apparatuses. Once defined, they can be used to efficiently describe both well-formed formulas and proofs within a formal system.

Let $U$ be the universe of discourse and $F_0\subseteq U$, called the foundation of the proof scheme. When applied to well-formed formulas (wffs), the foundation will be the set of atomic wffs. When applied to formal systems, the foundation will be the set of axioms of that formal system.

A transformation rule $R$ is an $n$-ary function in $\left[H\to U\right]$ where $H\subseteq U^n$. By $\left[H\to U\right]$, we mean the set of all functions whose domain is $H$ and range is $U$. Let $R$ be the set of transformation rules for the proof scheme.

A proof is a finite sequence $langle{x}_1,\mathrm{...},x_{n}rangle$ of elements of $U$ such that for all $k\in \left\{1,\mathrm{...},n\right\}$ we have either $x_k\in F_0$ or there is an $R\in R$ and a subsequence $langle{y}_1,\mathrm{...},y_{k-1}rangle\subseteq langle{x}_1,\mathrm{...},x_{n}rangle$ such that $x_k=R\left(y_1,\mathrm{...},y_{k-1}\right)$. If $langle{x}_1,\mathrm{...},x_{n}rangle$ is a proof then it is called a proof of $x_n$. $x$ is called provable if there is a proof $langle{x}_1,\mathrm{...},x_{n}rangle$ in which $x=x_n$. This is written $⊢x$ or $\varnothing ⊢x$.

A proof scheme can be specified by an ordered triple $\left(F_0,U,R\right)$ where $U$ is any set with $F_0\subseteq U$ and $R$ is a set of transformation rules. If $P=\left(F_0,U,R\right)$ is a proof scheme then we will let $T\left(P\right)=\left\{x\in U:⊢x\right\}$. Intuitively, $T\left(P\right)$ is the set of all tautologies of the proof scheme $P$.

Formal Systems

A formal system is a deduction apparatus in which one defines wffs over an alphabet and proves theorems from a set of axioms using some inference rules. More formally, a formal system $F$ is a 4-tuple $\left(A,G,T,I\right)$ where $A$ is a finite set (considered to be symbols), $S\left(A\right)$ is the set of all finite strings (also called utterances), $G\subseteq S\left(A\right)$ is the set of wffs (also called statements), $T\subseteq G$ is the set of axioms, and $I$ is the set of inference rules. $I$ is called an inference rule if there exists an $n\ge 1$ and there is an $H\subseteq G^n$ such that $I\in \left[H\to G\right]$.

Sometimes one defines $G$ by specifying a set of atomic wffs $G_0$ and a set of transformation rules $R$ that enable us to “build” new wffs from old wffs. We then can say that $G=T\left(G_0,S\left(A\right),R\right)$, considering the proof scheme $\left(G_0,S\left(A\right),R\right)$. When specifying a formal system this way, a formal system can be said to be a 5-tuple $\left(A,G_0,R,T,I\right)$ where $G_0\subseteq S\left(A\right)$ is a set of atomic wffs and $R$ is the set of formation rules (which are viewed as transformation rules) for wffs, $T\subseteq G$ is a set of axioms, and $I$ is a set of inference rules.

A proof in a formal system is a finite sequence $langle{x}_1,\mathrm{...},x_{n}rangle$ of elements of $G$ such that for all $k\in \left\{1,\mathrm{...},n\right\}$ we have either $x_k\in T$ (i.e., $x_k$ is an axiom) or there is an $I\in I$ and a subsequence $langle{y}_1,\mathrm{...},y_{k-1}rangle\subseteq langle{x}_1,\mathrm{...},x_{n}rangle$ such that $x_k=I\left(y_1,\mathrm{...},y_{k-1}\right)$. If $langle{x}_1,\mathrm{...},x_{n}rangle$ is a proof then it is called a proof of $x_n$. $x$ is called provable if there is a proof $langle{x}_1,\mathrm{...},x_{n}rangle$ in which $x=x_n$. This is written $⊢x$ or $\varnothing ⊢x$. If $M\subseteq G$ and there is a finite sequence $langle{x}_1,\mathrm{...},x_{n}rangle$ of elements of $G$ such that for all $k\in \left\{1,\mathrm{...},n\right\}$ we have either $x_k\in M\cup T$ or there is an $I\in I$ and a subsequence $langle{y}_1,\mathrm{...},y_{k-1}rangle\subseteq langle{x}_1,\mathrm{...},x_{n}rangle$ such that $x_k=I\left(y_1,\mathrm{...},y_{k-1}\right)$, then we say $M⊢x_n$.

Here are the first few digits of pi in base 10
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467678374494482553797747268471040475346462080466842590694912933136770289891521047521620569660240580381501935112533824300355876402474964732639141992726042699227967823547816360093417216412199245863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818347977535663698074265425278625518184175746728909777727938000816470600161452491921732172147723501414419735685481613611573525521334757418494684385233239073941433345477624168625189835694855620992192221842725502542568876717904946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886269456042419652850222106611863067442786220391949450471237137869609563643719172874677646575739624138908658326459958133904780275900994657640789512694683983525957098258226205224894077267194782684826014769909026401363944374553050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382686838689427741559918559252459539594310499725246808459872736446958486538367362226260991246080512438843904512441365497627807977156914359977001296160894416948685558484063534220722258284886481584

Pi is traditionally defined to be the ratio of circumference to diameter in any circle. In the taxicab space which forms a rectangular grid of possible routes the taxi can take (streets), the distance between (0,0) and (1,1) is 2 (a whole number). As the crow flies, the Pythagorean theorem would tell us that distance is the length of the shortest route between (0,0) and (1,1) is the square root of 2 (an irrational number).

Pi is involved in what many say is the most beautiful of equations, attributed to Euler:
e^(i Pi) + 1 = 0

You can probably change the definition of Pi to be the only real number between 3 and 4 that is a solution to this equation:
e^(i x) + 1 = 0

I wonder if anything special would come of defining Pi that way.

There are various set theories that have a universal set (which would be absolutely infinite)...one involves using 3-valued logic.

The thing to research regarding this is the axiom of comprehension, as it is called.

If set theory A has a completely unrestricted axiom of comprehension, set theory A will automatically then have a universal set, namely the A-set consisting of all A-sets that equal themselves.

For the axiom of comprehension is a powerful way to "generate" new X-sets from the other X-sets known to exist.  The general idea is that a set can be defined in an unambiguous way as being the ensembles all of whose members share a common property.  For example, the description "integers having remainder 2 when divided by 3" "generates" a set as described.

If we take an unrestricted axiom of comprehension, then it states something along these lines:
For every description there is a set whose elements fit that description.
(Hmm, I wonder if "grammatical systems" would help here.)

Seems innocent enough, right?  Well Russell came along and showed that this unrestricted axiom of comprehension, devised by Cantor, leads to a set theory in which every statement is both true and false; a set theory in which every statement is both true and false is probably not going to be considered interesting at all.

Russell came up with a particularly clever and brutal disproof of the axiom of unrestricted comprehension that Cantor believed was true.  This, some speculate, is why Cantor finished his days in an asylum because, deep down, he had lost his faith and lost his way by even acknowledging the possibility that God does not exist.  He thought set theory was a mathematical description of God as the ultimate set so big it contains all sets.

If anthropomorphizing God (God to man) is tricky and/or wrong, then going in reverse (man to God) is probably equally as uncertain.  Cantor was trying to do this with his set theory, imho.

If only Cantor had lived a bit longer.  His successors eventually came up with different set theories in which there is an ultimate set so big it contains all sets.

To Cantor it might have been his goal to prove that God exists but that argument which still happens a lot to this day often boil down to one's definition of the word God, or say undefined if not.

In NFU, the universal set is "generated" by an axiom and there is an axiom of restricted comprehension.  In a paper by Skolem, three-valued logic was used to reveal a set theory with an unrestricted axiom of comprehension was, in fact, a theorem!

I am not sure which of these set theories are stronger or weaker than the others (except in the obvious cases).

Imho, Cantor is one of the giants on whose shoulders his successors do rest.  I'd have to say he's one of my favorite mathematicians.

More on Grammatical Systems

Variations on a theme here. Some interesting and perhaps seemingly different things all share something in common... Music that can be written in sheet music, language that can be faithfully written, a huge chunk of math (if not all maybe), martial arts kata and choreographed dance, games like chess and baseball, and basically anything communicated by some symbols, all share some essential features. I think linguists have a name for these types of structures; I wrote earlier that I would call them Grammatical Systems. So if one can prove something about all grammatical systems, it would hold true for all the items in the list automatically. Trouble is I'm not sure how much I can prove about GS's. I did find one fact about GS's which isn't really deep but it's a start. Trying to find out if others worked in this already, linguists or philosophers of language so I don't have to re-invent the wheel. Well, we'll see where this goes...

Now what I think about a TOE is that it would be like blueprints for reality. I think I can explain it with an analogy: reality is to a TOE as music is to its sheet music. The latter being a sort of mathematical encoding of reality from which all of reality can be "recovered" if deciphered correctly. Looking at it this way, actually writing a TOE down seems pretty far off.

However, if there are blueprints to reality, reality's sheet music if you will, then that would make reality itself a grammatical system. I'd like to know if that's true or not. I doubt it but I feel it's worth investigating. The grammar would roughly speaking be the laws of physics.

A poem about everything

A poem about everything.


May you find your compass
use it to return home
don't close conduits of connections
God is the ONLY voice you hear
allow the water around you to send ripples
many out of one
one out of many
reverse, reverse, reverse
how am I going to be the only one who remembers
times before the physical was discovered
we led incorporeal lives
no possible way to describe a full portrait of God
because there is no need
and in no need, may the suffering cease
and in ceasing of suffering, you find
everything

This picture cold represent how a map of reality might look.  Here, one color can represent corporeal and the other color can represent the incorporeal.  Notice that both colors are shown to varying degrees of brightness.  This could mean that there are many types or levels of existence.  The two, corporeal and incorporeal, in some convoluted way form a confluence, that confluence being singular and the underlying principles behind the scenes are revealed. 

Proofs and Formal Systems

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formal systems and spacetime signatures

I wonder about spacetime signatures.


I don't think I've told anyone that I hypothesize that reality actually is a formal system, though what "formal system" would mean there might not be what it is elsewhere in literature so I would of course have to first define what I mean. 


Now, formal systems are very much like laboratories except these are mathematical laboratories.  In these laboratories, there must be four components:
1.  the symbols to be used (e.g., for communicating)


2.  a conglomerate of which utterances are considered in this laboratory to be grammatically correct


3. a conglomerate of which grammatically correct strings of symbols (i.e., statements) are axioms and assumed true in this laboratory


4.  a conglomerate of transformation rules that input so many statements (called premises) and output another set  of statements called conclusions (or the conclusion if there is only one statement in the output).  The transformation rules are called inference rules.


The only other related and relevant concept pertaining to formal systems is what could be called "the consequence closure of the formal system."  In order to discuss this, one needs to define what a proof is and what a theorem is.


A proof of a statement (i.e., grammatically correct utterances) is a finitely long sequence of statements such that, for all statements in the proof, that statement is either an axiom or the output of any of the transformation rules when applied to all previous statements in the proof.


A theorem is just any statement that is embeddable in a proof (which is a finite sequence of statements).  Intuitively, a theorem is true relative to the ambient formal system; within this formal system a theorem is merely a true statement (truth being relative here).


The consequence closure of a formal system is the ensemble of all theorems of that formal system.




Finally, back to spacetime signatures.  If reality is a formal system, then we must be among its theorems.  A parallel universe would just be any formal system embeddable within reality (viewed as a formal system).  Perhaps the consequence closure of that parallel universe is a good way to define its spacetime signature.