It’s a couple months old now (just like J. Gresham Machen when he was baptized), but be sure to go back and listen to this delightful Office Hours with Dr. Van Drunen (wait, did you think I meant the other Dr. Van Drunen?), for it’s fascinating look into the family and social history of the hero of our faith, J. Gresham Machen.

The interview covers a lot of ground, including Machen’s relationship with his mother (Minnie), his Southernness and perspective on race, a snapshot of early 20th century cultural change through Minnie’s diaries and correspondence, and Machen’s early childhood. For instance, I learned that Machen earned a whole dollar for memorizing the entire Westminster Shorter Catechism by age 6 — but didn’t even attend church with the family until he was 5!

For funsies, we can also have a guessing game. There’s a connection between Machen and myself (and OHS DGH! (and I could go on and on)) — whoever can guess it wins a prize ($10 worth of bragging rights and moral superiority).

[Update: The winner is Paul, who correctly guessed that Machen & Hart & I were all edumacated at **The** John**s** Hopkins University, the seal of which I lovingly reproduce to the right.]

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Does it have to do with the state of PA?

I’d rate your guess as: not exactly “cold” (compared to the passing winter), but cool enough to remind one that summer is coming and make you put on a jacket.

Second born?

greatly disliked by fundamentalists?

Interesting guess. I don’t know about DGH, but I’m a firstborn.

What, are you saying fundamentalists don’t like me?

Note: I don’t want to mislead, I don’t recall the connection being mentioned in that interview. I learned about the connection a few months ago, and the interview just reminded me about Machen.

Now that’s a picture I want a copy of- where did you find that one? And the baby looks so intent and focused on the text. Who says infants need video stimulation?

I found it by searching images.google.com for “baby catechism“, and swiped it from here.

This image would also have been good…

(I just remembered, I once blogged about firstbornness)

Does it have to do with geography?

Nope, Machen is a Baltimorean, I’m from California (originally GRusalem), and I don’t know about Hart. I have the impression Philly, but that’s probably just because of his association with WTS.

I know the connection.

Thanks for waiting, since you have a genetic advantage. I was kinda hoping DGH would pop in and guess it, but it’s probably enough to have this mystery revealed “on the third day”…

I would’ve made a guess, but I was hoping for a few grand worth in Table-Talk points. Sniff.

Being the Official Calvinist Blogger of Table Talk radio, I’m pretty sure that deputizes me to award Table Talk radio points…

Machen taught at Princeton, in

New Jersey.Hart has published with P&R,

New Jersey.Rube got a doctorate in math at Rutgers,

New Jersey.All three went to

Johns Hopkinsat some point.All three were/are members of the

OPC.I would say all three know their math, but Hart once told me with a straight face that there are no necessary mathematical truths since you can add in different bases. He thinks 2+2 yields a different sum depending on the base you add in.

And I think Machen would have liked me. I believe Rube likes me. Hart wishes I were dead, though; so I guess that one’s out too. Anyway, I took some swings, did I land any punches?

Ding ding ding!

I had almost forgotten this puzzle was hanging out there, but JHU was the connection I was aiming at. JG and I were undergrad, Hart was PhD.

Yikes! DGH should stick to history, and let Poythress cover the math. There’s a difference between what a number

is, and how it is represented. Is arabic 2 different than roman II? Or consider one of my favorite jokes: “There are 10 kinds of people, those that know binary, and those that don’t”. That joke only works in print, which is why I keep it on my homepage (and thanks to my in-laws, I even have it on a t-shirt! Teh geek chic!)Another of my favorite jokes: There are three kinds of people; those can can count, and those that can’t.

(I guess DGH would be the third kind?)

“Yikes! DGH should stick to history, and let Poythress cover the math.”

The problem isn’t history, it’s his desire to make sure Christianity has *nothing* to say to these secular (common) activities. It’s the problem I’ve tagged as “worldview minimalism” that led to his problem.

BTW, what do you think of Poythress’ thoughts on math?

Double BTW, I’m reading a book on the philosophy of math by Stewart Shapiro. It’s very good, you may like to check it out at some point.

I scanned Poythress’ article, and found it interesting, but not particularly useful. He is not really talking about math itself, but the philosophy behind math. He claims different mathematical results stemming from different mathematical philosophies, but I’m not convinced. Like he talks about various geometries (Euclidean and non-Euclidean), but they don’t really achieve different results. In Euclidean geometry, nonparallel lines always intersect at exactly one point. In Spherical geometry, nonparallel lines always intersect at exactly two points. But that’s not to say that they disagree with each other, or one (or both) of them must be “wrong”, they come out with different theorems because they have different definitions of “point” and “line”, etc.

I didn’t know you had a doc in math, Rube. That’s really impressive! I’ve heard math is good stuff when involved with music. Must by why you’re such a good violinist!

Betcha didn’t know I do to! My work is entirely self-guided and my status was self-awarded. All my studies are focused on failing to appreciate or even comprehend anything more complicated than closing out a cash register. LoL

Well thanks! I try not to talk about my doc-ness all the time, because usually it is not relevant. I’m sure your self-doc is just as relevant for most things!

Hmmm, while I’m not a huge fan of Poythress’ stuff on the matter, I think he’s right insofar as postulate V is false since it was taken to be a *necessary* truth, which means that it supposedly holds in all possible worlds, including non-Euclidean geometry. Granted, all is fine if one confines oneself to the world of Euclid (which we know is false, assuming a realist view of math and physics), but the problem is that the theorems were thought to be necessary truths (Kant called V a synthetic a priori), and V (at least) clearly isn’t. Thoughts?

pV isn’t “false” simply because Euclidean geometry is insufficient to describe our physical universe. Euclidean geometry is what you get if you assume postulates I-V. To the extent that we talk about physical ‘points’ and ‘lines’ and ‘angles’ and ‘polygons’ and ‘circles’ in a ‘space’ that all conform to Euclidean definitions of those terms, then correctly-derived Euclidean results will hold.

Thus I wouldn’t call Euclidean Theorems “necessary truths”, but rather I would call them “necessary consequences of the Euclidean postulates”. Same as for different but seemingly similar theorems in non-Euclidean geometries.

Consider 2D “geometry” (I use scare quotes to highlight the false assumption that there is only one valid 2D geometry). If the space you have to work with is flat, and extends infinitely in all directions, then you have pV and Euclidean geometry. Given a line and a point off the line, there is exactly one parallel line through the point. All triangles have angles that sum to

exactly180 degrees.But if the space you have to work with is the surface of a sphere, then a line (what you get when you join two points and extend in both directions) is a great circle, and for every line AB, and point C not on AB, every line through C intersects AB exactly twice. Every triangle has angles that sum to

less than180 degrees.BUT, if you constrain yourself to a local area of that sphere, geometry behaves only neglibly different than the Euclidean 2D geometry in a plane tangent to the sphere in that local area. Is the difference important? It depends on how accurately you can measure. Draw a circle — what is the area? Well, which area — the area of the circle bounded by the inner-width of the pencil mark, or the outer? When you are building a schoolhouse, does it matter that if you walk long enough in the same direction, you return to your starting point? Does it matter that your walls are not truly “parallel” because the extension of their lines intersect on the back side of the earth? ((a) no, and (b) by pouring and grading a concrete foundation, you establish a local Euclidean geometry anyways)

Which geometry is “true”? They’re both true, inasmuch as they follow from their definitions and axioms (necessarily), and they’re both useful, if you can find a physical (or abstract) situation that conforms to the definitions and axioms.

Rube,

To the extent that we talk about physical ‘points’ and ‘lines’ and ‘angles’ and ‘polygons’ and ‘circles’ in a ‘space’ that all conform to Euclidean definitions of those terms, then correctly-derived Euclidean results will hold.But I-IV hold in spherical geometry. You would think if it was a matter of definitions, this wouldn’t be the case.

“Thus I wouldn’t call Euclidean Theorems “necessary truths”, but rather I would call them “necessary consequences of the Euclidean postulates”. Same as for different but seemingly similar theorems in non-Euclidean geometries.”But that’s what everyone else called them, which is why they claim V is false.

Moreover, they can’t be “necessary consequences” since V isn’t a consequence of I-IV in non-Euclidean geometry, yet those hold. Hence, there’s a circumstance where the premises are all true and yet V doesn’t follow.

A necessary consequence says that the conclusion *always* follows from the premises, i.e., there are no possible worlds where it doesn’t—for that’s just what necessity means. So perhaps you’re using ‘necessary’ in some special sense, if so, how are you using it?

Oh, and I don’t think “true” is the right word to use if you claim that both follow simply within their systems and are both useful. Being useful doesn’t imply being true. Phlogiston was quite useful for a time, but we now know it wasn’t a true entity.

Paul, I hope you can hear me from way down here in my knotted position, but I think you’re mistaking Christian secularism with legal secularism (and calling it “worldview minimalism”). LSism is actually what wants to make sure Christianity has nothing to say to secular or common activities, not CSism. But it’s no wonder you do so, since:

“…Christian and legal secularists will likely agree or disagree on public policy or legislation or electoral candidates on the basis of political philosophy or gut instincts about public life, not on the basis of belief or non-belief.”

Contra LSism, CSism does believe that faith has some bearing on public life, but not in the direct way the theocrats or theonomists or transformers do. I get that you think Christian secularism is heinous, but you should at least understand that it isn’t saying what you think it is.

V isn’t a consequence of I-IV; Euclidean geometry is a consequence of I-V (where “2D space” is defined as flat and infinite). Spherical geometry is a consequence of space being defined as the surface of a sphere, along with analogous redefinitions of ‘point’, ‘line’, etc, which give I-IV new meanings, and for which V is not true, but V’=”…exactly two intersections”.

Or (inside out) if you simply postulate I-IV+V, then planar geometry is what fits “underneath”; provides a working model for visualizing the abstract propositions. Likewise, if you postulate I-IV+V’, then spherical geometry is what fits “underneath”. (I don’t know, there may also be other visualizations that might “fit underneath”)

Zrim, no, I am not making such a mistake. Indeed, if you read what I wrote (I know that’s hard for you), you’d note the parenthetical after ‘secular,’ which shows it was being used in the Christian and not secular sense.

Now, I note you’re trying to sound more sensible, given the years of my arguing with you that Christianity does indeed have some things to say to all of life, and I appreciate your coming around. But the original meme was that the Bible had “nothing” to say to “epistemology, math, politics, ethics, law, science, etc.,” and now that’s moved to “something.” Remember, Hart saying on his blog that there was no such thing as “a biblical epistemology?” Remember him saying that there’s no such thing as a Christian worldview? Remember you claiming the same thing? However, since you admit it does have some things to say, then there are “Christian” views of those things. But this was another term you and Hart bristled against for quite some time. You know have to make room for a “Christian worldview” and a “Christian view of science.” For the longest time this language was not permitted. Finally after years of arguing, you’re admitting you’ve been wrong. See, I always knew you were overstating your case, and this is what I told you years ago. Imagine the progress you could have made if you would have listened to me back then. Well, better late than never, I suppose. Glad to see you seeing things this way.

So now we’ll see no more Zrim quotes like:

“Some might be tempted to counter the neo-Calvinist “all of life” dogma by saying that Scripture doesn’t speak about common tasks but it does speak to them.

But the problem still is that the Bible doesn’t really speak either about common tasks or to common tasks.It actually speaks to God’s covenant people who do common tasks. To say that the Bible speaks to a common task seems to be the first step to saying that there really are redemptive versions of whatever creational task.”I can pull quotes like that, and worse ones, all day long.

OK then, self-consistent. Or, let

P: For any triangle, the sum of the interior angles is 180 degrees

p: For any triangle, the sum of the interior angles is less than 180 degrees

(I-V–>P) is true.

(I-V’–>p) is true.

But is P true, or is p true? Such questions are meaningless until you define ‘triangle’, ‘interior angle’, ‘degree’, etc. (i.e. specify which geometric axioms you are operating with)

Neither can we simply ask “are I-V ‘true'”? (Which is I guess what you are pointing out)

Rube, this is interesting. It looks like you take a constructivist view of math, whereas I’m a mathematical realist. Anyway, I’ll leave it at this since it’s off topic. But maybe we’ll pick it up via email or something,

I’ll just say in the history of math, many have tried to show that V is a consequence of I-IV (google it). There were reasons for this, given what non-Euclidean geometry showed. If V was an entailment of I-IV, it could be dispensed with.

Second, V was taken to be true and its logic form is represented with a universal quantifier. Indeed, for quite some time it was taken to be a necessary truth. Now, when it was shown to not hold on spherical geometry, many took this as showing that V was false (since it had been taken to be a necessary truth). If the philosophy of math gets redefined to talk not about truth but about constructions and self-consistency, then you can say that V hasn’t been shown to be false, but you lose with it the notion of mathematical truths as truths about the human mind independent world. So either way, the matter has a lot of bearing on the phil. of math, and to that extent, Poythress is right to point out the issues with V.

Lastly, I’d just like to get your take on why I-IV hold on a sphere but V doesn’t. Why would this be if the reason V doesn’t is because of how it defined points and lines given that I-IV uses those same terms and defines them in the same way V does?

OK.

I don’t know if “constructivist” is bad, but I don’t believe that math is “not really ‘out there'” or “only in our minds”.

Constructivist isn’t bad, it’s actually a pretty tough position to refute. I just think it’s false. You say you don’t believe math is a construction of human minds, but some of the things you say imply this.

I bet that if the abstract propositions I-IV were to be concretized with algebraic definitions in the R^2 plane (or in some other way make the planarity assumptions explicit), V could be derived from I-IV (plus planarity concretizations). But I’m just guessing.

Spherical may be able to use the same literal axioms I-IV, using the same terms, but they are not defined the same way. For instance, “straight” has to change its meaning (unless you can you gradually levitate and walk into space?). And “lines” are quite different since you can follow a line infinitely, but in planar you always get further from where you started, but in spherical you will always cycle back over where you started.

Anyways, I think I’ll take it offline now…

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