reperiendi

Fractional topological dimension

Posted in Math by Mike Stay on 2007 July 27

We say a topological space X has topological dimension m if every covering C of X has a refinement C' in which every point of X occurs in at most m+1 sets in C', and m is the smallest such.

Any point x is clearly in at least one set; it’s the other sets that x might be in that concern us. The dimension m turns up in a sum over sets in the refined covering C'. If we want to move to groupoid cardinality, then the sum should be over equivalence classes of sets in C'.

Let C'_{(x,A)} be the set \{A' \in C', A' \ne A, x \in A' \}. Given (x,A), we need some way to get a group G_{(x,A)} acting on C'_{(x,A)}. If we have such a group, then we can say the topological dimension is the maximum over (x,A) of the groupoid cardinalities of C'_{(x,A)} // G_{(x,A)}.

Any idea how to get such a group?

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Theocosmology

Posted in Theocosmology by Mike Stay on 2007 July 26

The brain is an amazing thing; hypnosis can block pain. There are plenty of hypnosis weight-loss clinics for helping people to lose weight and to quit smoking. As we get more detailed knowledge of the brain, it will become a trivial thing to improve one’s willpower. Instead of procrastinating, we’ll be able to tell ourselves to do something and become compulsive about doing it. No more late homework!

NPR’s science Friday has several programs on memory; in one recent one (that I can’t find), they explained that memories are recreated every time we remember them, with an accompanying degradation of the memory. Details we don’t think are important are discarded and filled with plausible reconstructions; the sudden recall of something you haven’t thought of for twenty years is far closer to the actual event than a memory that you think about all the time. It’s also easy to introduce fake memories, because the brain fills in plausible details in exactly the same way.

They know this because of recent experiments using drugs that interfere with the formation of memory: rather than have the patients try to learn something, and watch how they forget it, they had them try to remember something, which they proceeded to forget even after the drug wore off. Humans now have the abillity to selectively erase conscious memories.

There’s another form of memory that’s subconscious, of habits and feelings acquired through repetition. This seems to be a deeper, more protected form of memory. Clive Wearing, who was once a musicologist, contracted encephalitis, which damaged his hippocampus. He has a seven-second memory. He’s constantly feeling as if he’s just awoken from a dream, having no idea where he is or what has happened to him for the past twenty years. But he can still direct his choir perfectly and fills with joy every time he sees his wife.

It’s probable that we’ll be able to erase or modify those memories in the near future, too.

We admire people who do things despite pain. Words like courage, tenacity, and kindness all apply to enduring in the presence of pain. The atonement is the prototype here: love overcoming pain.

The whole concept of mortality in scripture is predicated on the idea that pain is necessary for progression; it separates souls like a chromatograph separates particle sizes. People who are physically strong become so by exercising and enduring pain. Smart people typically work hard to become so. Barbie was right: math is hard–but at the moment there’s no other way to become so.

If we can remove the memory of pain and decide to remove the pain that stands between us and some desired end, will there be any virtue in the accomplishment? Skousen’s argument for the purpose of the pain in the atonement was to work on the compassion of intelligences. Where does compassion go when we can forget the hurt at will?

Negative dimensional objects and groupoid cardinality

Posted in Category theory, Math by Mike Stay on 2007 July 26

I was thinking about some stuff involving fractals and non-positive-real dimension. It’s still a very rough idea, though.

There’s the concept of topological dimension, which is necessarily an integer. It looks like it’s typically the floor of the Minkowski dimension.

One way of talking about iterated function systems is to consider patterns of digits in n-ary strings. For instance, the typical Cantor set consists of those points in the unit interval whose ternary expansion contains only 0’s and 2’s. It’s a single coordinate, but restricted in the values it can take.

We can add the dimension of two vector spaces by taking the categorical product = direct sum \oplus. We can multiply the dimension by taking the tensor product \otimes. We can add and multiply the dimension of two unit hypercubes with the same construction.

It’s clear how to tensor integer-dimension objects; I was looking at how one might tensor fractional-dimension objects. I have an example with the Cantor set that probably generalizes.

Consider the set of points C in the unit interval whose 4-ary expansion consists exclusively of the digits 0 and 3:

----------------
----        ----
-  -        -  -

etc.

C has dimension \log_4(2)=1/2. C \otimes C is isomorphic to the set of points where pairs of digits in base 4 are either 00 or 33—which is the same as saying that in base 16, each digit is either either 0 or F=15. The dimension of this set is \log_{16}(2)=1/4. So as before, tensoring two objects multiplies their dimension.

Now consider \displaystyle D=\prod_{i=0}^{\infty}C^{\otimes i}

  • C^{\otimes 0} is the unit interval.
  • C^{\otimes 0}\times C^{\otimes 1}= 1\times C is a subset of the unit square, a set of parallel stripes. The dimension of 1\times C is 1+|C|+|C|=1.5
  • C^{\otimes 0}\times C^{\otimes 1}\times C^{\otimes 2}= 1\times C\times C\otimes C is a subset of the unit cube, consisting of parallel copies of the previous set. The dimension of 1\times C\times C\otimes C is 1+|C|+|C|^2=1.75

and so on until

  • \displaystyle D=\prod_{i=0}^{\infty}C^{\otimes i} is a subset of an infinite-dimensional hypercube with dimension \displaystyle \sum_{i=0}^{\infty} |C|^i = 1/(1-|C|) = 2

John Baez introduced something called groupoid cardinality. We can add and multiply finite sets using the disjoint sum and cartesian product. We can “divide” sets by using the weak quotient of a set by a group:

\displaystyle |S//G| = \sum_{[x]} \frac{1}{|\mbox{Aut}(x)|},

where [x] is an equivalence class and x is equivalent to y if they’re in the same orbit.

So consider the set 3=\{a,b,c\} under the action of \mathbb{Z}_2 via reflection:

\leftarrow | \rightarrow

a\, b\, c

That is, 0 in \mathbb{Z}_2 maps each point to itself, while 1 in \mathbb{Z}_2 maps b to itself and swaps a and c. So a and c are in the same orbit and form one equivalence class. The automorphism group of a is only the identity. The orbit of b includes only itself, so |\rm{Aut}(b)| = |\mathbb{Z}_2| = 2. This forms a groupoid: a category where all morphisms are isomorphisms.

Summing over equivalence classes, we get

\displaystyle |3//Z_2| = 1/|\mbox{Aut}(a)| + 1/|\mbox{Aut}(b)| = 1/1 + 1/2 = 3/2.

Is there some way to view the dimension of the Cantor set as arising from some kind of weak quotient?

He also talked about something he calls “structure types”, which is very much like a generating function, but acts on groupoids (with sets as a special case) instead of on numbers. So in the structure type

\displaystyle \frac{a_0}{0!}x^0 + \frac{a_1}{1!} x^1 + \frac{a_2}{2!} x^2 + \ldots,

the term a_n/n! really means a_n//S_n, where a_n is the set of structures you can put on an n-element finite set (like, say, the set of binary trees with n nodes) and S_n is the group of permutations of n objects. The term x^n is the cartesian product of n copies of the groupoid x.

It apparently makes sense to talk about groupoids with negative cardinality as well as fractional cardinality. For example, taking x to be the groupoid 3//Z_2 described above, the cardinality of the infinte groupoid

\displaystyle 1 + x + x^2 + \ldots=\frac{1}{1-x}=\frac{1}{1-3/2}

is -2!

This suggested to me that we could do something similar and construct a negative-dimensional object. If we stick in a square, we get a set

line \times square \times 4-cube \times 8-cube \times 16-cube \times \ldots

This doesn’t make much sense as an isolated set: all we can say is that it’s an infinite-dimensional hypercube. But if we had some way of knowing that the product we were taking was over 2^n-cubes, then we could use the formula above.

Here’s the idea: if restricting the values a digit can take in an n-ary expansion reduces the dimension, then expanding the values ought to increase it. Since this doesn’t make much sense with points in the real number line, let’s move to \mathbb{N}[[x]] instead: let C be the set of Taylor series in x with coefficients in \{0,2\}. We could say C has dimension \log_n(2) when x=1/n. (I still have to work out how the topology changes with x.)

Then we can define a set T of series with, say, dimension 2, and construct an infinite product

\displaystyle \prod_{i=0}^{\infty} T^{\otimes i}.

This object ought to have dimension -1; at least, it will satisfy the property that multiplying the dimension by 2 (by tensoring with a square) and adding one to the dimension (by multiplying by a line) will give the original object: if d is the dimension, then d satisfies 2d+1=d.

The structure type for binary trees evaluated at the trivial one-element groupoid gives an infinite groupoid with cardinality \frac{1+i \sqrt{3}}{2}, so complex cardinalities (and therefore complex-dimensional objects) can also arise.

Can we make this rigorous? Is there some obvious connection to the complex dimensions of fractal strings?

Theocosmology

Posted in Theocosmology by Mike Stay on 2007 July 23

The View from the Center of the Universe says we’re the first generation that can tell stories about the creation that might actually be true. We should take the mythological imagery we have around us and reinterpret it so that it corresponds to something real instead of something imagined, thus giving people a sence of place in the universe.

Their stories, by most accounts, weren’t very good. But I believe they’re on the right track: stories are how humans understand the world. It’s part of why I like John Baez’ teaching style so much: he tells stories. A search on his site for the word “story” turns up hundreds of occurrences!

Gödel, Escher, Bach talks about string rewrite rules and theorem proving, trying to illustrate the difference between syntax (structure) and semantics (meaning). He states some rules like

xpyqz => sxpyqsz

and then suggests that “sx” is the “successor” of “x”, and thus one interpretation of “p” and “q” is “plus” and “equals,” respectively. But it could just as easily be “equals” and “taken from.” How do we know what a structure “really means?” Does that question even make sense?

Universal algebra is a language for describing simple structures like sets, monoids, groups, rings, and modules (but not fields). Lawvere showed that every algebraic theory corresponds to a cartesian category, and that a functor from the cartesian category to Set gives a model of the theory. The Yoneda lemma says that there’s a one-to-one correspondence between the set of models of a theory and the set of functors out of the theory. So any model has to arise as a functor out of the theory.

This lets us separate syntax from semantics, structure from meaning. The theory is a Platonic ideal that has models in the “real world” of sets and functions. Of course, now that the “real world” has been revealed to be quantum, we should consider functors into other categories, too: a functor into Hilb assigns quantum meaning to the structure. This is how Feynman diagrams are used for quantum field theory: every line represents a Hilbert space of states instead of a set, and each vertex represents a linear operator instead of a function.

Usually, what a structure “means” can only really be answered by considering all of the possible meanings together and looking at the relationships among them. But that’s a very hard task in most cases. It’s easier to have a single, typical instance in mind: what do you think of when I say “undirected graph?” It’s probably an image of a not-so-special graph, not the mathematical definition.

Choosing some example is essential to the way we learn. Adam and Christ, the “last Adam,” are the prototypes used by the Gospel. But there are a lot of details left out.

I’m trying, in these Theocosmology posts, to build up a backstory for my existence, which is the story I’m most interested in; I’m choosing details that may or may not be true, but are as true-to-life, as typical as I know how to make them.

Theocosmology

Posted in Theocosmology by Mike Stay on 2007 July 13

Joy, glory, and knowledge

Here’s the passage that refers to element and spirit:

D&C 93:33-34 For man is spirit. The elements are eternal, and spirit and element, inseparably connected, receive a fulness of joy; and when separated, man cannot receive a fulness of joy.

In the Book of Mormon, joy is used almost interchangeably with glory. Nephi says

2 Ne 33:6 I glory in plainness; I glory in truth; I glory in my Jesus, for he hath redeemed my soul from hell.

C.S. Lewis’ sermon The Weight of Glory discussed an apparent paradox:

I turn next to the idea of glory. There is no getting away from the fact that this idea is very prominent in the New Testament and in early Christian writings. Salvation is constantly associated with palms, crowns, white robes, thrones, and splendour like the sun and stars. All this makes no immediate appeal to me at all, and in that respect I fancy I am a typical modern. Glory suggests two ideas to me, of which one seems wicked and the other ridiculous. Either glory means to me fame, or it means luminosity. As for the first, since to be famous means to be better known than other people, the desire for fame appears to me as a competitive passion and therefore of hell rather than heaven. As for the second, who wishes to become a kind of living electric light bulb? (ibid., p. 5)

He concludes that it is the joy of a child in finding favor with his father, being “famous with God.” Being known of God as opposed to the opposite:

In some sense, as dark to the intellect as it is unendurable to the feelings, we can be both banished from the presence of Him who is present everywhere and erased from the knowledge of Him who knows all. We can be left utterly and absolutely outside—repelled, exiled, estranged, finally and unspeakably ignored. (ibid., p. 7)

Spencer W. Kimball said, “Real intelligence is the creative use of knowledge” (CR, Oct. 1968, p. 130. Emphasis mine.)

Hofstadter examined creativity in his book Metamagical Themas (http://tinyurl.com/29v7fq). He talks about Knuth’s Metafont package: in that program, a font has lots of different properties that can be adjusted. There’s an enormous space of letters that can be expressed by these parameters, but they are nowhere close to encapsulating all the forms we recognize as, say, the letter ‘A.’ Coming up with a new parameter, for Hofstadter, is creativity–and yet, ironically, such creation must arise exclusively as the result of some deterministic process. He implies he could not have written any other book than the one he did–but within the book, a few pages away from this claim, he talks about how we continually conceive of worlds slightly different than our own, such as one where the state line between Illinois and Indiana is a few miles to the west of where it is here.

In the section on parquet deformations, he mentions the “flickering” effect of looking at a Necker cube. My brother Doug suggests that agency and qualia are part of the same mystery. We have no idea how perception of shape or color arises in the mind, nor how independent action arises: but we can choose to see the cube with its top toward us or away from us.

In Moses, God says that

Moses 1:39 …This is my work and my glory—to bring to pass the immortality and eternal life of man.

In other words, what God prefers to spend his time on is creating more gods. Even Richard Dawkins couldn’t take offense at that doctrine. God is an adult of the species, a father in the literal sense. What can we guess about producing minds that last forever?

Greg Egan has treated artificial minds and immortality in several of his stories. In Orphanogenesis, artificial minds are run on a computer, and when they become self-aware, the computer revokes all authority for the mind to be modified from the outside; the consequence is that some minds descend into eternal madness. A rejection of truth is a rejection of reality. My guess is that a faulty view of the universe leads to eternal madness–outer darkness–if left unchecked, so in mercy, God checks it.

Was there ever a time or a place where an atonement couldn’t reach? Where all the minds were destroyed? Nephi writes about how we’d share Satan’s fate if not for the atonement in 2 Ne 9. There was clearly fear among the premortal spirits contemplating mortality (Rev. 5:3).