Comments on “The collapse of rational certainty”

Problems with infinities

Amos Ng 2020-04-12

Thanks for making this book public! I’m glad that you’ve written an interesting summary of what this section is to be, even though it’s not complete yet.

Could you say what the “ problems with infinities” you mention here are, so that I can look them up myself? I assume you aren’t talking about Gödel’s incompleteness theorem?

Any additional pointers to resources on the historical ramifications of such revolutions in mathematics would also be appreciated.

Thanks! No worries if it’s too much trouble :)

Russell's paradox, for example

David Chapman 2020-04-12

Hi, Amos,

This Wikipedia article section (on the “foundational crisis”) may be a helpful brief overview.

In the late 1800s and early 1900s, several problems showed up in the theory of infinite sets pioneered by Georg Cantor. One was Russell’s Paradox. The Wikipedia article I linked there gives a good explanation of why this caused so much trouble. Another was the continuum hypothesis.

These didn’t lead directly to Gödel’s and Turing’s no-go theorems, but those were part of the same general milieu (the Hilbert Program).

Enjoy reading about this! Apparently many great mathematicians found it awful at the time, but in retrospect it seems just really interesting.

For an overview of the crisis

frank 2020-05-05

For an overview of the crisis in mathematics that non-math folk can understand, I’d recommend linking the extremely good graphic novel Logicomix, which follows Bertrand Russell as he tries to uncover foundational . As someone who’s pretty mediocre at maths, this work was a really good introduction to the topic, the art is really good.

Quoting Saruman

Michael Taft 2021-01-02

The dwarves delved too greedily and too deep. You know what they awoke in the darkness of Khazad-dum… shadow and flame.

Awakening the Balrog

David Chapman 2021-01-02

Lol… I shall illustrate this with a suitable image from DeviantArt when I finish writing it!

Proofs too complex to check

SusanC 2022-01-24

In Plato’s Meno, “Even a slave can reason”.
For Martin Luther’s Protestant Reformation, anyone could read the scriptures.

This falls apart when the proofs get too big and complex to be checked by the average person in the street. (Or indeed, the average slave of Plato’s Athens).

Your proofs might be so big and complicated no-one can read them, (cf. the four colour conjecture). Sure, computers can check them, but people can’t.

If computers are our replacement for Plato’s slaves, we find ourselves in an unfortunate example of the master slave dialectic. computers can understand the proofs, but their masters can’t.

This puts a dent in Plato’s project of a rational democracy, where the citizens understand the proofs.

You’re instead in the realm of “this bronze idol” (sorry, I meant computer) “says the four colour theorem is true, so it is true. Why is it true? No idea.”

This destroys

Art and

SusanC 2022-01-24

Art critics sometimes point out reasons to doubt the popularly supposed connection between general relativity and Picasso’s Cubism. (Also in your article).

But on the other hand, compare Anselm Kiefer’s “The Ramanujan Summation, -1/12”. It’s a bit Cezanne like, in a Mont Sainte Victoire after a nuclear apocalypse kind of way, and picture space really is based on renormalization in quantum field theory/string theory. (Yep, that’s the Riemann Zeta function marked out in dead trees in the snow),

The reals

SusanC 2022-01-24

” It turned out that most ordinary, finite numbers are actually hard shells that encapsulate an actual infinity crammed inside under enormous pressure”

The reals? I don’t believe they exist! (Cf. The Princess Bride, and Kronecket’s God created the integers, all else is the work of man).

Seriously, that sentence threw me for a second. I was expecting an “ordinary finite number” to be an integer, not some bizarre thing like a real.

Calling them “real” doesn’t help either. It’s what they’re called. It doesn’t mean they’re like, actually real.

The reals

Ian Horswill 2022-01-24

Yes, I was expecting that sentence to lead to the construction of the natural numbers as equivalence classes of sets of the same cardinality, a la the Principia. Although now that I think about it, under their construction those are actual finite sets.

In any case, thank you for writing your books, David. I’ve enjoyed them very much.

How much did theoretical uncertainties really matter historically?

Brian Slesinsky 2022-01-24

It seems reasonable to assume that the Protestant Reformation really did result in hundreds of years of war, but how do we know that uncertainties about science and math had much to do with modern social upheavals? We can point to a lot of people saying it did, but a bunch of hippies saying “it’s all relative, man” doesn’t seem like quite enough to show that they were passionate about theoretical issues?

It seems like the important causes of that era were things like civil rights and opposing the draft and environmentalism and feminism and free speech. These are causes that people made significant sacrifices for. Or consider Marxism and labor issues. Or religion in the Middle East or Northern Ireland.

The chattering classes latching onto scientific and mathematical controversies as possible explanations for unrest doesn’t seem like quite enough to know that this is what the fighting was really about? Maybe this is giving academic controversies more power than they really deserve?

Cantor vs Kronecker

SusanC 2022-01-25

On reflection, I should have gone with “The Sets of Unusual Size? I don”t think they exist!”


Kenny 2022-01-25

I’m commenting here today to notify you that you’ve received a “public service medal or something”!

Platonic vs Leibnitzian proof

SusanC 2022-01-25

“Some on the left said that nothing should be true unless everyone could understand it, so relativity was elitist and anti-democratic and must be stopped.”

The way you phrase it is a bit facetious, but Plato’s idea of proof leads us to this point.

More carefully: nothing is socially accepted as true unless everyone can understand the proof.

Is a proof that’s too complicated for most people to understand still a proof? Under one serious philosophical candidate for what “proof” means, no, it isn’t a proof.

And yes, it is very anti-democratic. It completely wrecks Plato’s plan for how a democratic society can function. Oops.

Koide formula

SusanC 2022-01-26

I have just learned about the Koide formula. Within the limits of experimental error, the masses of the electron, muon and tau satisfy a simple arithmetic relation. But … no-one knows why. It’s not a prediction of the Standard Model. (This is following on from your aside about quantum theory being a rat’s nest)

The cat meme above is awesome and funny

Younes ben amara 2022-01-26

I am an avid reader of yours and as many people -as I’d assume- you are totally hit a home when you wrote:

That’s because no one has ever written this up before. Which is really extremely odd.

The emphasis mine.

I like the way you reveal the uncertainty of rationalism and in the same time not dismissing it all together.

However I agree with the commenter Brian Slesinsky regarding that the foundational crisis of rationalism and science is just a factor (and maybe even not the main one) among multiple factors that allow the hippies and other anti rationalism movements to extist, revive and thrive.

As you said in your email you do deserve a 🏅

I think your readers are certain that this piece of though is a gem.

Replying to several comments at once!

David Chapman 2022-01-26

SusanC — thank you for various interesting tidbits!

Ian — very nice to see you here! Glad you are enjoying. Hope things are well with you.

Brian — these are serious questions that I thought about quite a lot while writing, and don’t know how to get definite answers. (I alluded to that in this footnote.) How do you quantify the influence of one cultural event on another a few decades later? Maybe professional historians have ways of doing this; I know almost nothing about their methods. All I have is a bunch of first-person accounts from participants, and brief summaries from historians. How much should those be weighted?

That said, there’s a confusion here:

the important causes of that era were things like civil rights and opposing the draft and environmentalism and feminism and free speech

The New Left movement of the 1960s was not the same as the hippie/New Age movement. There was some overlap, but also quite a lot of antagonism. The New Left thought that the hippies were selfishly ignoring social problems to retreat into drug-addled spiritual fantasies, sexual hedonism, and pop-culture distractions. Those were, I’m confident, driven to a significant extent by anti-rationalism (because they were really explicit about it at the time).

I don’t think the collapse of scientific rationalism had anything to do with the rise of the New Left. I suspect (but don’t have good evidence) that it has quite a lot to do with the collapse of the US establishment in the 1970s.

Kenny, Younes — thank you both very much for the medals! :)


SusanC 2022-01-28

Way back when I was an undergraduate and had to sit through lectures on the concept of mathematical proof in Plato - most of which I have probably forgotten - there was some mention of the idea of proof implied by the Greek word δείκνυμι, to make evident.

Which word probably covers human comprehensible proofs, like most of the ones you encounter in Euclid, but probably doesn’t cover some enormously long proof that can only be checked with a computer…

vague pejoratives

joe grub 2022-01-28

Houston we have a problem.
I sensed intimations or strategic ambiguities in your line
… combined with popular pseudoscientific resistance to even sensible public health policies....
Would you mind defining better /
popular .
As in like majority/minority opinion as delineated by what media source.
as in like which scientific opinion expressed in which media source
as if there is one science with one data set?
as in defined by which “authority” on what is sensible/
So I can better get a grasp on what you are saying here

Lie Algebras

SusanC 2024-02-07

I have just, suddenly, had one of those moments where mathematics seems bizarre and crazy rather than logical and rational.

It all started when out dear St Dr Rev made a joke about the classification of Lie agebras.

Now, to be clear, I know nothing about Lie algebras other than (a) they were invented by some Norwegian guy called Sophus Lie; and (b) the mention of Lie algebras marked the exact point where I decided to bail on the General Relative course in undergrad. I am sure Ruth Gregory meant well, but I am out of here.

Anyway, as someone who knows zero about Lie algebra other than the above, I read Rev saying you of course want a classification of all such algebraic objects that are in some sense simple.

Me, who knows nothing, thinks: and then you discover some stupidly complicated example, which is then conjectured to be related to the Monster Group from group theory. To be clear, this was a joke.

I then look it up … ok, so you take the 8 dimensional E8 lattice you got from the classification of Lie groups, take triples when each element of the triple is from the E8 lattice, apply a further constraint to get a 3*8=24 dimensional sphere packing known as the Leech lattice, ask yourself what the automorphism group of this lattice is, and you get the Conway group…

ok, so stupid joke is in fact true.

But wait .. some versions of string theory are in 24 dimensional spacee. But .. but .. this would not initially appear to have anything to do with the E8 Lie algebra or sporadic groups. It just comes from the Riemman Zeta function. So, you assume there is some kind of high energy cutoff on excited states, because otherwise your equation would have infinity on both sides and that would suck (renomalisation). Maybe the exact value and shape of the cutoff doesn’t matter .. in which case we can imagine it looks like the Zeta function, just to make the math easier, which gives us a factor of 12, which we are subsequently going to finesse away by postulation that our theory of quantum mechanical, special relativistic elastic bands takes place in 24 dimensional space. No E8, at least not yet.

yeah, ok, so I have some dim memory of the Monstrous Moonshine conjecture being proved by Borcherds (never having dared look at the proof). Bur the Riemann Zeta function? Come on, this is just unreasonable.

String theory, 25 dimensions

SusanC 2024-02-07

(The above was very loose and off the cuff .. IIRC, string theory is in 25 dimensions + time, because one of the spatial dimensions is along the string, leaving you with 24 dimensions orthogonal to the string for E8 Lie Algebra/Monster Group/Riemann Zeta function craziness.

The anthropic principle is something like, the constants of physics do enable life to be possible, because otherwise we would not be here to be arguing about it, even if this would seem a piori unlikely.

I am getting a sense of a different principle, along the lines that the stupidest answer often turns out to be the correct one.

Like, suppose that the Creator of the Universe could pick anything from the Atlas of Finite Simple Groups when designing the laws of physics. But which one did they pick? Go on, guess…

Fortunately, not my problem

David Chapman 2024-02-07

I am so glad not to have to think about string theory :)

String Theory

SusanC 2024-02-08

I merely offer E8 * E8 heterotic string theory ( as an example of how an initially rational inquiry can leave you feeling like you agreed to spend the night in a haunted house to win a bet, and are now regretting the decision (cf. The House on Haunted Hill, dir. William Malone, 1999).

Call of Cthulhu

SusanC 2024-02-08

I feel like a character in the Call of Cthulhu role-playing game…

“Hey, I could do a bit where I pretend Cartan’s classification of Lie algebras has something to do with the classification of finite simple groups!”

[reads book, looses SAN points]

“Hey, the dimensionality of the Leech lattice just happens to be the same as that of the transverse oscillations of the string in string theory. No connection of course, that would be totally ridiculous. has nothing to do with Lie algebras, surely. But I could do a bit here.”

[reads another book, looses more SAN points]