[FRIAM] Excellent Beauty

[Frank Wimberly]

David,

I will address one of the Bear questions.  The rationals are a dense subset
of the real line.  That is, given any interval of the real line there
exists a rational number in it and it easily follows that there are a
countably infinite set of rationals in the interval (see below for a proof
found with Google.  I don't quite remember the proof.  It's been 55
years).  The rational numbers are however a set of measure zero.  See
another proof below.  That's equivalent to saying that choosing a real
number arbitrarily you have zero probability of choosing a rational.
(Nobody does this).  These are easy proofs because they could be much more
general.  If you have any questions, let me know.

Theorem:  The rationals are dense in the real line

Proof:  *Proof. *Since [image: x < y], we know [image: (y-x)> 0].
Therefore, there exists an [image: n \in \mathbb{Z}^+] such that

  [image: \[ n(y-x) > 1 \quad \implies \quad ny > nx+1. \]]

We also know (I.3.12, Exercise #4
<http://stumblingrobot.com/2015/06/30/prove-that-any-real-number-lies-between-exactly-one-pair-of-consecutive-integers/>)
that there exists [image: m \in \mathbb{Z}] such that [image: m \leq nx <
m+1]. Putting these together we have,

  [image: \begin{align*} nx < m+1 \leq nx+1 < ny &\implies \ nx < m+1 < ny
\\ &\implies \ x < \frac{m+1}{n} < y. \end{align*}]

Since [image: m,n \in \mathbb{Z}] we have [image: \frac{m+1}{n} \in
\mathbb{Q}]. Hence, letting [image: r = \frac{m+1}{n}] we have found [image:
r \in \mathbb{Q}] such that

  [image: \[ x < r < y. \]]

This then guarantees infinitely many such rationals since we can just
replace [image: y] by [image: r] (and note that [image: \mathbb{Q}
\subseteq \mathbb{R}]) and apply the theorem again to find [image: r_1 \in
\mathbb{Q}] such that [image: x < r_1 < r]. Repeating this process we
obtain infinitely many such rationals

Theorem:  The rationals are a set of measure zero.

Proof:  Date: 05/13/2001 at 08:18:54

From: Doctor Paul
Subject: Re: Countable sets and measure zero

if S is countable, then we can write down the elements of S:

     S = {s1, s2, s3, ...}

Now recall what it means for a set to have measure zero. It means that
given any epsilon > 0, we can cover S with a countable number of
intervals, rectangles, cubes, or "boxes" (depending on whether we're
talking about R^1, R^2, R^3, or R^k for k > 3) that satisfies this
property:

   the sum of the "volumes" (use length or area if appropriate) of
   these "boxes" is less than epsilon.

Since we're in R^1, we need to cover S with a countable number of
intervals such that the sum of the lengths of the intervals is less
than epsilon.

It's not an obvious proof, but once you see it, it's easy.

I'm going to use e for epsilon.

Put a disk around s1 of radius e/4. So you have essentially put a disk
that covers the interval {s1 - e/8 , s1 + e/8).

Now go to s2. Put a disk around s2 of radius e/8: you get
(s2 - e/16 , s2 + e/16)

Do you see the pattern?

When it's all said and done, you have an infinite number of intervals
whose lengths are:

     e/4, e/8, e/16, e/32, ...

We want to add these up and show that the total is less than epsilon.

By summing an infinite geometric series, we see that the sum of the
lengths of these intevals is e/2 < e, and that completes the proof.

This shows, for instance, that the rationals in the intervals [0,1]
has measure zero.



On Wed, May 27, 2020 at 2:37 PM Prof David West <profwest at fastmail.fm>
wrote:

> I recommended this book, Excellent Beauty by Eric Dietrich last week at
> vFRIAM. I think it has a lot to contribute to many of our recent
> discussions, but doubt that many will have/take the time to read it, so a
> short synopsis and discussion starting points might be in order?
>
> The book is fundamentally about "mysteries," things we do not understand,
> and two approaches to dealing with mystery: religion and science.
>
> Circa Galileo, The Church was the sole source of authoritative explanation
> and if science had the temerity to challenge that authority, either with
> contradictory explanations or exposure of mysteries for which The Church
> had no explanations, then science was heresy and suppressed.
>
> Fast forward and today Science claims to be the sole source of
> authoritative explanation. If religion has the temerity to challenge that
> authority then religion is derided, denigrated, and dismissed.
>
> The royal road, in fact the only road, to Truth is the scientific method
> and the method advanced by Peirce - who is quoted in the book.
>
> All would be well except for the annoying fact that roughly 80% of human
> beings remain religious and seek religion based explanations and often
> prefer the religion-based explanations over the scientific ones. Exemplar
> case: evolution.
>
> Which presents an interesting mystery — why does religion persist in the
> face of all the evidence against it? Can science offer and explanation for
> this mystery?
>
> Friend Jochen offered FRIAM his book, Hidden Genes, which addresses this
> issue. Jochen ground his argument in analogs / metaphors derived from
> genetics and evolution. Dietrich also looks to evolution. Parallels can be
> found between both arguments, but I will summarize Dietrich.
>
> Religion persists because it offers evolutionary advantage at the group
> level. Dietrich is borrowing David Sloan Wilson's arguments in Darwin's
> Cathedral; religion-as-group-glue.
>
> [[ Dietrich makes an error in this discussion, one that Jochen and Sloan
> make as well. Quoting Dietrich, "Note that all people on earth have a
> religion. No culture has ever existed that had no religion at all."  This
> is a mistake.  There is evidence that every culture, historic or pre, had
> some kind of belief in the supernatural.  Religion, however, is a modern
> invention, made possible only after agriculture and urbanization made it
> possible for people to adopt specialist roles: butcher, baker, candlestick
> maker, PRIEST, KING, theologian, etc. Religion has existed for about the
> last 3-4000 years, out of 1.3-1.8 million years of human existence. ]]
>
> Dietrich borrows Daniel Dennet's explanation for the origin or religion:
> humans have a series of "devices" for facial recognition, agent detection,
> "unusual but memorable combinations" (e.g. talking trees, Thor's hammer),
> explanation, and, combining the preceding in various ways, a fiction
> generating device. The latter explains religion. Having these devices
> enhanced individual survival and fiction==>religion provided the group
> selection advantage.
>
> All interesting, but still leaves open the question of from what source
> should we seek explanations of mystery: religion or science.
>
> Gould proposed "non-overlapping magesteria" as a compromise. Dietrich
> demolishes this suggestion, leaving science as the only explainer standing.
>
> But, there are mysteries that are beyond science — the "excellent
> beauties." An excellent beauty is a mystery that science reveals but cannot
> explain. And Dietrich means CANNOT, not cannot yet.
>
> Three specific excellent beauties are discussed and a half-dozen others
> are introduced. The first of which is "The Conscious Self." This is a topic
> that has come up often at FRIAM and those discussion, I believe, would be
> greatly enhanced with a reading of this section of Dietrich's book.
>
> A second example is "Infinity and Beyond" which I will not even attempt to
> summarize, but I encourage all the mathematicians on the list to read what
> he says and explain it to me.  BTW, mathematicians, did "Godel derive a
> solution to Einstein's relativity equations that shows time is circular"?
> Can you explain that to a bear of little brain?
>
> The third discussion is of the "Rarity of the Commonplace" — how the
> mundane reality about us is so improbable. Part of this discussion deals
> with causality (another frequent FRIAM topic, but not so much recently),
> part with the fact that only 4% of the Universe is ordinary matter, and
> part that is corollary to the discussion of infinities. Part of the latter:
> a Real Line is made up of the rational and the irrational numbers, both of
> which are infinite in number. The rationals are infinitely dense and so
> take up zero percent of the Real Line.  Again, mathematicians, a bear of
> little brain asks for an explanation.
>
> The conclusion of the book is an argument for why excellent beauties are
> beyond scientific explanation.
>
> He juxtaposes Chalmers notion of "scrutability" (comprehensibility) —
> which is held to be the exemplar of the Enlightenment / Peircian notion of
> science — and Robert Heinlein's (from the book Stranger in a Strange Land)
> "grok."
>
> Chalmers' scrutability thesis: "there is a compact class of truths such
> that knowing those truths suffices to know all other truths about our
> universe."
>
> Grok: thorough, intuitive, empathic understanding.
>
> E.g., we comprehend quantum physics to a robust extent, but we do not grok
> it.
>
> Science can expose profound mysteries. Those mysteries are natural not
> supernatural and religion is of no use in exploring or explaining them,
> Enlightenment ideals of science (and Peircian method) offer means to
> explore the mysteries and will generate, as side effects, lots of
> scrutability, but will not, can not, result in groking those mysteries.
>
> ********
> I would really like to discuss many of the ideas in the book, but need
> others to take the time to read at least some parts of the book. Especially
> chapter ten and the appendix to chapter ten.
>
> davew
>
>
>
>
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-- 
Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918
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