The Trinity

Some people seem to think that the Christian doctrine of the Trinity is illogical. This article, for instance, when telling of the Council of Nicaea's affirmation that Jesus is God and of one substance with the Father, says "At Nicaea, the radicals were the orthodox, and theirs was a defiant stand against the idol of reason, the false deity of logic". Sometimes people use the words "reason" and "logic" to express what is actually impressions, imagination or intuition. That might be happening in this article; I see phrases like "it makes more sense" and "sounds reasonable". If that's what they mean by "logical", then maybe the doctrine of the Trinity is illogical. But if "logical" has to do with real logic — truth relations between propositions that express predicates, relations and suchlike — then I'm going to say no.

Is 3 = 1 is illogical? Yes, but 3 and 1 of what? The doctrine of the Trinity is the proposition that God is three persons that have one essence. The careful reader will notice that it's three persons and one essence, and persons and essences are not identical. There's no problem saying there are three persons in one family, four quarts in a gallon, or seven days in a week, so what's the problem? I see two possiblities:

• One possibility is that some people assume that a god is, by definition, a person. God is a person and God is three persons: 1 = 3. All I can say to that is that was never part of the Christian idea of God, and it's obviously not part the Council of Nicaea's idea of God.
• The other possibility is that we understand the doctrine of the Trinity to mean God is one being and three persons, and some people also make the assumption that each person is a distinct being. God is three persons and one being; each person is a being; therefore, God is three beings and one being. But is it really true that all persons are distinct beings?

What makes us consider two things to be different "beings"? It can't be simply because we can make a distinction between them. An electron's mass and an electron's electrical charge are distinct, but they are properties of the same being — the electron. In other words, the word "being" implies something more than distinguishability from other things. It can't be existence. An electron's mass exists. An electron's charge exists. Be we don't call them "beings". If we say that the difference between masses and charges and persons is that mass and charge are properties, but persons are beings, we beg the question. What is the difference between a property and a being? Why can't a person be a property? Considering each human person to be a distinct being makes sense because humans are in every way separable from each other. We each have our own minds. Each of us occupies a different location in space. We can be separated by time. One human individual's existence doesn't necessarily entail another human's existence. I can imagine a world where Abraham Lincoln existed, but I didn't exist. I can imagine a world where Abraham Lincoln existed, but Mary Todd Lincoln didn't exist. I can even imagine a world where Abraham Lincoln existed, but his mother didn't exist. (Surely, Abraham Lincoln must have had a mother, but not necessarily the one that he did have.) But that doesn't work for the Father, the Son and the Holy Spirit. Their existences necessarily require each other, because the relationships between each other are part of each of their definitions. They can't be separated by time, because they are coeternal. They can't be separated by space, because God (or God the Father, at least) is everywhere. That leaves their minds: It seems clear to me from the thoughts that Jesus expressed in the Gospels that he and the Father have their own minds, but I don't see how that amounts to them being separate beings. They seem to have different thoughts, but ultimately, they never disagree, and Jesus never opposes the Father.

Possible Objections

• What decides a person's essential properties? Why is Jesus being Yahweh's son necessary, but Abraham Lincoln being Nancy Lincoln's son not necessary? Also, could there be something arbitrary about how we divide the universe into beings? When we say that one series of sets of molecules defined across time is me, and another is Abraham Lincoln, is it just convention or convenience, and not the expression of some metaphysical reality? Maybe. But that doesn't make the idea of God being three persons and one being self-contradictory. And if there are no metaphysical realities to consider when defining beings and essential properties, then the idea can't be false. The worst we can say is that it is difficult or complicated. But if there really is an all-powerful creator of the universe, what makes you think that such a being wouldn't be complicated or difficult to understand?
• There's a chance that someone might read this and think that I'm committing the heresy of partialism. Partialism (according to Christians who object to partialism, at least; I don't know anyone who is themselves a partialist) is the idea that Father, Son and Holy Spirit are fully God only when they exist together. Sort of like how Captain Planet appears only "by your powers combined", I guess. But that's not what I'm saying. I'm saying they are eternally coexistent. An electron exists where a certain mass and a certain charge are combined into one thing, and mass and charge can exist in other things, but for Father, Son and Holy Spirit, there are no "other things". There is only the Trinity, and the Trinity has always existed and will always exist.

Bailout Monopoly

Here are some changes to the rules of Monopoly that I think will make the game more fun.  The first two are not new. Everyone I know already plays with them:

1. All required payments to the bank for taxes, Chance and Community Chest go to the center of the board.
2. When a player lands on Free Parking, they get all of the money on the center of the board.

The next five changes are new:

3.  Any amount of money a player is required to pay for rent, tax, Chance, or a Community Chest, that they can't pay with their own cash or with money gotten by mortgaging property, is paid by the bank.
4. A player always receives the rents for their property, including additional rent for houses and hotels, and doubled rents for monopolies, even if their property is mortgaged.
5. A player does not have to sell houses or hotels back to the bank before they can mortgage their property.
6. All money paid, when purchasing properties, houses and hotels from the bank, goes to the center of the board, instead of the bank.
7. The game ends when the bank runs out of money.  Whoever has the most assets - cash plus property plus houses plus hotels minus mortgage debt - wins.

The last change is also new, but I am not sure if it is necessary yet:

8. Some of the money in the bank - maybe a few thousand dollars - is removed at the beginning of the game so the bank runs out of money sooner.  This would prevent the game from going on for too long, or from one player accumulating enough money that it is clear that the other players can't win.

 Apart from these changes, the game should be played with the standard rules, as written.  If a player owes money and has unmortgaged property, they must pay as much of the debt as they can by mortgaging property, but Rule 3 implies that a player doesn't have to sell houses and hotel back to the bank.  Players may buy, sell or trade properties amongst each other.  A player may not build houses on a property unless all properties of the same color are not mortgaged.  Once all houses and hotels have been purchased from the bank, no more may be bought.

Because of Rule 3, I'm calling Monopoly with these rules "Bailout Monopoly".  Probably the worst part of traditional Monopoly is the slow, inexorable loss of assets that you experience when you are losing.  Rule 3 prevents that from happening.  If you can't pay, the bank bails you out.  Also, no one can go bankrupt, so everyone gets to stay in the game until the end.

The most fun part of Monopoly, in my opinion, is collecting rent - especially when you have a monopoly and houses.  With Rule 4, you can always collect rent on your property, and because of Rule 6, you don't have to give up houses once you've built them.

Rules 1, 2, 6 mean that a lot of money can be gotten by chance.  That, and the luck normally involved in landing on properties, will make it impossible, at least in some games, to predict who will end up with the most money, and therefore who will win, until the end of the game.

There you have it.  If you try it and like it, maybe leave a comment.  Or leave a comment if you don't like it, or if you think of an improvement.

A Way to Apply Bayes' Rule to Conflicting Evidence

Here's an application of Bayes' Theorem that, IMO, deserves to be more widely known:

where A is a set of possibilities, E is a set of sources of evidence, and i_e is the event that a source of evidence indicated whichever possibility it indicated (as opposed to indicating some other possibility).  It's really a general formula for deriving probability from any number of conflicting, corroborating, and possibly biased sources of evidence.  So long as the sources of evidence are independent and they each indicate exactly one of a limited number of possibilities, this formula applies.  For instance, suppose you have several people making claims about something that happened, some agreeing and some disagreeing.  How do you weigh their testimony against each other?  If you can find a way to estimate how likely each person was to make the claim that he or she claimed given each of the possibilities, this formula provides you with a way forward.

Here's a math paper that states the formula as a theorem and proves it: Bayes' Theorem Under Conditional Independence.  It's Theorem 4.  (The paper expresses the formula with different symbols, but it's essentially the same thing.)  I haven't found it anywhere else in print or online. Since the numerator on the right-hand side is the basis for naive Bayes classifiers, maybe it's included in some AI course material somewhere.  I suppose it falls between two stools; it isn't one of the most basic applications of Bayes' theorem, so it wouldn't be in introductory material, but to a professional mathematician, it might seem too obvious to deserve mentioning.

Professor Kirke's House

The filming location for Professor Kirke's house in the BBC version of "The Lion, the Witch and the Wardrobe" (IMDb) is Llanvihangel Court.  Some of the rooms, especially the spare room that contains the wardrobe, look like they might be sets, but everything else appears to be filmed there.  For evidence, watch this YouTube copy of "The Lion, the Witch and the Wardrobe", specifically at these times, noting:

• 3:16 - the front of the house,
• 23:10 - the fountain,
• 48:31 - the garden house (guardhouse) next to a red tree and
• 49:01 - the stained-glass window,

and compare them to these 2017 photographs of Llanvihangel Court and the photo of the stained-glass window here: Elizabeth I from Elizabeth I and the Family of Charles I - work from Stained Glass in Wales (llgc.org.uk)

Unlike some other locations for the series, the house wasn't mentioned in closing credits.  At the time of filming, it was a private house and the owners may have not wanted publicity.  In recent years, the house appears to have hosted tours and events.  It has a web site: https://www.llanvihangelcourt.com/

I found the house by looking for recognizable objects in scenes set inside and outside the house and searching the Internet for them.  It took me a while to notice the stained-glass window, but eventually I did, and that was the key.  It depicts a lady that looks like Queen Elizabeth I.  On closer inspection, I saw that there is a gentleman on either side of the lady and one of them is laying down a cloak.  I recalled there was an incident where Sir Walter Raleigh laid down his cloak for Queen Elizabeth so she would not have to step in a puddle.  Googling "elizabeth", "cloak", "raleigh", and "stained glass" as required terms turned up the "Stained Glass in Wales" web page mentioned above, which identifies the location of the stained-glass window: Llanvihangel Court. 

As for the puddle incident, Wikipedia says it is probably apocryphal.  Another source flatly states that it never happened because the first known written reference to it was written 80 years later by a Thomas Fuller, who sometimes fabricated stories.  Of course, it is a non sequitur to go from "this person sometimes fabricates stories" and "this person was the first person we know of to write about it" to "we know this never happened", so I disagree.  You can read Thomas Fuller' account of the incident here.  Actually, it isn't so much an account as a mention.  Quoting Fuller: "his introduction into court is said to have born an elder date, from the time he spread his plush cloak for Her Majesty to step upon over a wet place".  It seems like Fuller is stating hearsay or repeating tradition.  As it is written, I don't know why he would have made it up himself.  For myself, I would rate the incident to be about as likely to be true as any bit of remembered information about a historical person, or any other unproven statement made in the book it was included in.

"New Symbols for Base-16 and Base-256 Numerals" Bested on Reddit

In 2017, I co-authored a paper that presented a system of hexadecimal digits that could be combined as ligatures to form base-256 digits.  It's online here.  In 2019, someone posted a system of base-256 digits on Reddit that I and the other author agree is better: Made a surprisingly simple base-256 numerical system...

In our paper, we enumerated 9 desirable qualities for a set of hexadecimal digits in order to evaluate different sets.   This other system has/hasn't the qualities as follows:

• MNE: Yes
• STR: No
• LIG: Yes
• AMB: Yes
• DSP: Yes
• BIN: Yes
• 0: No
• 1: No
• TRN: Yes

This system has 6 of the 9 enumerated qualities.  This is 2 fewer qualities than our published system, but I nevertheless think it is a better system because 1) its base-256 "ligatures" are easier to write and 2) I can remember how these digits encode binary numbers much more easily than our published system, and therefore this system would be easier for me to use.  Well done, u/Tuckertcs, whoever you are.

On another topic, in our paper we suggested the name "sedecisedecimal" for base-256 numbers.  I still think it's the best Latin/Greek-based name for them.  Would anyone seriously prefer "duocentehexaquinquagesimal"?

Bill Watterson's Political Cartoons

While attending Kenyon College, Bill Watterson (the author of Calvin and Hobbes) was editorial cartoonist for the school newspaper, The Kenyon Collegian.  Most weeks during his tenure, they printed a cartoon of his, usually a political cartoon.  Since the Kenyon Collegian's archives are available online, you can see them all.  The following is a complete set of links to issues of the Kenyon Collegian that have these cartoons: 

• Volume CIV (1976-1977), issues 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27
• Volume CV (1977-1978), issues 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26
• Volume CVI (1978-1979), issues 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24
• Volume CVII (1979-1980), issues 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27

It's Rational to Believe in Jesus

So says Aron Wall (and me).  He has posted a fair amount of material on his web site that explains how believing what the Bible says about Jesus can be rational.  Several of those postings amount to as good a case for Christianity as I've seen expressed anywhere, so I'm posting links to them for your benefit (and my own; I will re-read them).  They are listed in an order that I think makes sense for them to be read if they were to be read in sequence.

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There is good evidence that Jesus was resurrected from the dead and that he is the Messiah:

1. Can Religion Be Based on Evidence?
2. Theology: Less Speculative than Quantum Gravity
3. Christianity Is Based on Observations
4. Let Us Calculate
5. Christianity Is True

The Old Testament part of the Bible is not a fabrication:

• Some Comments on Biblical History

Christianity is more likely to be true than other religions:

1. Introduction
2. World Evangelism
3. Ancient Roots
4. Supernatural Claims
5. Historical Accounts
6. Early Sources

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Note: I don't know Aron Wall personally; I've just read his blog and agree with his approach to Christian apologetics.